Rethinking entanglement of a single particle

Dr. Bee has caused me to rethink entanglement, and reader Ajit sends a paper on Entanglement isn't just for spin

Quantum entanglement occurs not just in discrete systems such as spins, but also in the spatial wave functions of systems with more than one degree of freedom.

It is sometimes said that Einstein discovered entanglement in 1935, and it was immediately recognized as the central defining feature of quantum mechanics. But as the above paper notes, the word was not in common use until about 1987, and did not find its way into textbooks until after that.

As the article explains, entanglement is not some peculiarity of tricky spin experiments. It is a property of all quantum systems.

Entanglement is explained as the thing that makes quantum mechanics nonlocal, and hence the essence of why the theory is non-classical and mysterious.

Paul Dirac one said:

Quantum-mechanically, an interference pattern occurs due to quantum interference of the wavefunction of a photon. The wavefunction of a single photon only interferes with itself. Different photons (for example from different atoms) do not interfere.

This is not an exact quote, but he said something similar.

This is a confusing statement, and I would not take it too literally. But in a similar spirit, I would say that a quantum particle can be entangled with itself.

Entanglement is often introduced by describing creation of a pair of particles with equal and opposite spins. But it is much more common. In any atom with several orbital electrons, those electrons are entangled. Nearby particles usually are. The case of the equal and opposite pair is interesting because that gives distant entanglement, but nearby entanglement occurs all the time.

Consider a stream of particles being fired into a double slit. Each particle is interfering with itself, and is entangled with itself. The interference results in the interference pattern on the screen.

The entanglement results in each particle hitting the screen exactly once. If you purely followed the probabilities, there are many places on the screen where the particle might hit. Those possibilities are entangled. If the particle is detected in one spot, it will not be detected in any other.

You cannot understand the experiment as localized probabilities in each spot of the screen.

Viewed this way, I am not sure the 2-particle entanglement story is any more mysterious than the 1-particle story. Maybe explanations of entanglement should just stick to the 1-particle story, as the essence of the matter.

President Joe Biden is Politicizing Science

Lawrence Krauss has a WSJ article attacking Pres. Biden for politicizing science.

The New Scientific Method: Identity Politics
The National Academy of Sciences fights bias by explicitly introducing more of it.

Lubos Motl praises the article.

In particular there is now an aggressive affirmative action program at the National Academy of Sciences, where less competent women and Blacks are being appointed in order to meet diversity quotas.

Biden's White House Coronavirus Response Coordinator is a Democrat political hack named Jeffrey Zients. Donald Trump had an immunology expert in that job.

The authorities are still not telling us the truth about the virus. See this article by a NY Times science reporter on evidence it came from the Wuhan Institute of Virology.

Update: From a CDC official in a press conference, as reported in the NY Times:

DR. WALENSKY: … There’s increasing data that suggests that most of transmission is happening indoors rather than outdoors; less than 10 percent of documented transmission, in many studies, have occurred outdoors.

The paper goes on to explain that the true number is more like 0.1%. Yes, that is less than 10%, but appears to be a distortion attempting to justify outside mask requirements.

Quantum wavefunction is not everything

Reader Ajit argues that I am not following textbook quantum mechanics properly. He has posted Postulates of quantum mechanics, as stated by various authors.

Checking other versions of the postultes, I find:

The state of a system is completely described by a wavefunction

Associated with any particle moving in a conservative field of force is a wave function which determines everything that can be known about the system.

I wonder why this would be stated as a postulate. It is not used by the theory anywhere, and it is not true.

Sometimes it is stated for a single particle, but it cannot be true if the particle is entangled with another. Sometimes it is stated for scalar wave functions, but that cannot be true if the particle has spin.

You can correct those problems by introducing spinor-valued wave functions of several variables, but then you are still ignoring quantum fields and all sorts of other complexities.

Now you might say: Okay, but if use the whole Standard Model, or some bigger unified field that takes into account all possible interactions, and then we construct a wave function of the universe, then that would completely describe the state of the universe.

That would not be quantum mechanics. That would be some theorist's fantasy that has never been carried out.

Quantum mechanics is a theory that takes in some available info, and makes some predictions, but never achieves a complete description of the system. Nobody has any idea how any such complete description would ever be accomplished.

Take a simple example, the Schroedinger Cat. The wavefunction is a superposition of dead and alive states. Is it a complete description of the state of the system? No, of course not. The cat is either dead or alive. You can get a more complete description by opening the door and looking to see if the cat is dead. The wavefunction is most emphatically not giving a complete description.

I don't know why anyone would say that the wavefunction is a complete description of the system. Other physics theories do not start off with a postulate declaring some sort of god-like omniscient. It doesn't make sense to even say something like that.

And yet this postulate is prominent on various lists of postulates for quantum mechanics. I will have to do some further research to find out who is responsible for this silly idea.

This week's Dr. Bee video is on Einstein's spooky action at a distance. She says that the spookiness is the measurement update (ie, collapse of the wavefunction), not entanglement.

Believing that the wave function is a complete description necessarily causes these spooky concerns. Any observation affect distant parts of the wavefunction. If the wavefunction is a complete physical thing, then it is spooky.

Does Quantum AI have Free Will?

A new paper argues that a quantum computer could be conscious, and have free will.

Since I am skeptical that quantum computer will ever achieve quantum supremacy, you probably think that I dismiss this as nuts. Actually I don't.

Turing machines are deterministic, and do not have free will. But I believe humans do. The mechanism is not understood, and may involve quantum mechanics. So maybe a quantum computer can do that, even if cannot factor large numbers.

The London Guardian has a good essay on the arguments about free will. It says:

Harris argues that if we fully grasped the case against free will, it would be difficult to hate other people: how can you hate someone you don’t blame for their actions? Yet love would survive largely unscathed, ...

I personally can’t claim to find the case against free will ultimately persuasive; it’s just at odds with too much else that seems obviously true about life.

I agree with that last sentence. A lot of intellectuals reject free will, but in the process they also reject a lot of things that seem obviously true.

I do not agree with the love/hate analysis. If I believe that someone has no free will, and is merely a preprogrammed robot to do evil things, then sure, that is a good reason to hate him. He would be a sub-human evil nuisance. A puppet of the devil. As for love, try telling your wife that you only love her because the chemicals in your body have made that illusion. Some psychologists say that, and I don't think it helps.

The article says that philosophers have gotten death threats over such issues.

Jerry Coyne endorses most of the essay, but argues:

Contracausal free will is the bedrock of Abrahamic religions, which of course have many adherents.

No. Islam doesn't accept free will. Moslems are always talking about God's will being carried out, as if no one can do anything about it. Jews have mixed views. Catholics believe strongly in free will, and so do many Protestants, but some, such as Calvinists, do not.

A previous Guardian essay by historian Yuval Noah Harari said:

Unfortunately, “free will” isn’t a scientific reality. It is a myth inherited from Christian theology. Theologians developed the idea of “free will” to explain why God is right to punish sinners for their bad choices and reward saints for their good choices. If our choices aren’t made freely, why should God punish or reward us for them? ...

You cannot decide what desires you have. You don’t decide to be introvert or extrovert, easy-going or anxious, gay or straight. Humans make choices – but they are never independent choices. ...

But now the belief in “free will” suddenly becomes dangerous. If governments and corporations succeed in hacking the human animal, the easiest people to manipulate will be those who believe in free will.

He is gay Israeli Jewish atheist. Perhaps he is a slave to his programming, but others are not.

Theologians did not invent free will. The Gospels use phrases like "go and sin no more". This assumes that you can choose to sin, or not sin.

You can decide to be an introvert or extravert. Change is not easy, but people do it.

No, the easiest to manipulate are those who think that they are already slaves.

Coyne argues:

Free will skepticism (sometimes called “hard determinism”). As you must know, this is the view to which I adhere. Though it’s often called “determinism”, with the implication that the laws of physics have already determined the entire future of the universe, including what you will do, that’s not my view. There is, if quantum mechanics be right, a fundamental form of indeterminism that is unpredictable, like when a given atom in a radioactive compound will decay. It’s unclear to what extent this fundamental unpredictability affects our actions or their predictability, but I’m sure it’s played some role in evolution (via mutation) or in the Big Bang (as Sean Carroll tells me). Thus I prefer to use the term “naturalism” rather than “determinism.” But, at any rate, fundamental quantum unpredictability cannot give us free will, for it has nothing to do with either “will” or “freedom”.

I call this argument: Only God has Free Will.

Coyne is an atheist, but he seems to believe in some sort of Spinoza God. Humans have no freedom or free will. We are just puppets being controlled. God is not a predictable robot, and can make choices for us and the world. God even guides evolution of biological species by directing mutations.

The phrase "fundamental quantum unpredictability" means that the human observer can only predict probabilities. It always leaves open the possibility that someone with more info could make a better prediction. If Coyne wants to believe that it is some sort of God making all our choices for us, I guess that possibility is allowed.

For example, a quantum mechanics textbook might say that a uranium atom has a certain probability of radioactive decay in the next hour. And maybe that is all that can be said with the info available. But nowhere will it say that it is impossible to make a better prediction, if the state of the atom could be more precisely determined. As a practical matter, it is hopeless to get the wavefunctions of all the quarks in a uranium nucleus, but the point remains that better info might give a better prediction.

In my opinion, attributing all the decisions in the world to a Spinoza God is contrary to common sense and experience, and does not really solve anything. It is like a turtle argument that atheists like to mock. In fact, I worry about the mental health of anyone who believes that, as it is similar to schizophrenics who say that they are obeying voices in their heads.

Feynman quote on Leftist Groupthink

...There was a special dinner at some point, and the head of the theology place, a very nice, very Jewish man, gave a speech. It was a good speech, and he was a very good speaker, so while it sounds crazy now, when I’m telling about it, at that time his main idea sounded completely obvious and true. He talked about the big differences in the welfare of various countries, which cause jealousy, which leads to conflict, and now that we have atomic weapons, any war and we’re doomed, so therefore the right way out is to strive for peace by making sure there are no great differences from place to place, and since we have so much in the United States, we should give up nearly everything to the other countries until we’re all even. Everybody was listening to this, and we were all full of sacrificial feeling, and all thinking we ought to do this. But I came back to my senses on the way home. The next day one of the guys in our group said, “I think that speech last night was so good that we should all endorse it, and it should be the summary of our conference.” I started to say that the idea of distributing everything evenly is based on a theory that there’s only X amount of stuff in the world, that somehow we took it away from the poorer countries in the first place, and therefore we should give it back to them. But this theory doesn’t take into account the real reason for the differences between countries—that is, the development of new techniques for growing food, the development of machinery to grow food and to do other things, and the fact that all this machinery requires the concentration of capital. It isn’t the stuff, but the power to make the stuff, that is important. But I realize now that these people were not in science; they didn’t understand it. They didn’t understand technology; they didn’t understand their time. The conference made me so nervous that a girl I knew in New York had to calm me down. “Look,” she said, “you’re shaking!..."

New book on Free Will Debate

Philosophers Dan Dennett and Greg Caruso wrote a a 2018 debate on free will, and now they have expanded it into a book. From a review, they agree more than they disagree:

Both are naturalists (JD p.171) who see no supernatural interference in the workings of the world. That leaves both men accepting general determinism in the universe (JD p.33), which simply means all events and behaviours have prior causes. Therefore, the libertarian version of free will is out. Any hope that humans can generate an uncaused action is deemed a “non-starter” by Gregg (JD p.41) and “panicky metaphysics” by Dan (JD p.53). Nonetheless, both agree that “determinism does not prevent you from making choices” (JD p.36), and some of those choices are hotly debated because of “the importance of morality” (JD p.104). Laws are written to define which choices are criminal offenses. But both acknowledge that “criminal behaviour is often the result of social determinants” (JD p.110) and “among human beings, many are extremely unlucky in their initial circumstances, to say nothing of the plights that befall them later in life” (JD p.111). Therefore “our current system of punishment is obscenely cruel and unjust” (JD p.113), and both share “concern for social justice and attention to the well-being of criminals” (JD p.131).

Their hair-splitting philosophical differences are not that interesting. What interests me is how they could both have such a screwed-up view of life, and still think that they are on opposite sides of a big issue.

Caruso says we have no free will. Dennett says that we think that we do, and it is useful to maintain the illusion, but it is not real.

Without free will, there ia no consciousness. Our systems of law, ethics, morality, and politics depend on free will. Christianity is based on it. So is the scientific method. It is hard to imagine how modern civilization could even exist without free will.

These philosophers discard it all based on a belief that all events have prior causes.

When a uranium nucleus decays, is it determined by prior causes? Our best scientists cannot answer this question. But somehow these philosophers can get the answer by playing silly word games? No, it is all nonsense.

Trans Ideology and the New Ptolemaism

The social sciences often make cosmological analogies, and screw them up so badly that I cannot even tell what point they are making.

Here is a new scholarly paper on an academic dispute:

Trans Ideology and the New Ptolemaism in the Academy ...

Ptolemy constructed an inordinately complex model of the universe in order to make all of the empirical data conform to a central, organizing false assumption, namely, that the earth was at the center.

Foucault’s influence in the academy is at least as often lamented as celebrated, and I will not attempt in what follows a comprehensive critique of his work. Instead, I will focus on one tendency his example has encouraged, which, using Rockhill’s analogy, I will call the “new Ptolemaism.” This is a push for scholarship to be insistently insular and to be much less interested in the study of the world than in the study of the study of the world. This kind of work, which is by now very common in the social sciences and humanities, performs the same neat trick every time. It turns out, in every such analysis, that the framing of inquiry turns out to be more significant than the object of inquiry. ... ...

Consider present day calls to remake the academy. There should be more soft sociology of the hard sciences; there should be more women in male dominated disciplines; we should “indigenize” the university. There are two terms in each case; we should reverse the conventional hierarchy of those terms; and the results will be profoundly liberatory, because, Ptolemaically, the university rather than the world is the most important locus of struggle. ...

Gender critical feminists like me notice, of course, that one infinitely more often sees and hears the slogan “transwomen are women” than its counterpart “transmen are men.” To understand why this is the case, you’d have to pay attention to patterns of power in the world rather than to Ptolemaic valence-flipping. One of the signs on my office door that most infuriated feminist academic women colleagues on social media described the parallels between men’s rights activism and trans rights activism. Many feminist academic women clearly saw it as their moral and intellectual duty to decry this assertion.

The Foucealt here has nothing to do with the Foucault pendulum, which helped prove Ptolemy wrong about the motion of the Earth. No, it is a French post-modernist and pedophile rapist.

The author might have some valid points about feminism and trans ideology, but the Ptolemy stuff is nonsense, and the Foucealt stuff probably is also.

Ptolemy did not construct an inordinately complex model of the universe. It was not any more complicated that any other model achieving similar accuracy. He did assume that the Earth was at the center, but the model is not really any different or more complicated from that. He descibed the stars, Sun, planets, and Moon as seen from Earth, so he would have to include the calculations needed for that whether the Earth moved or not. It was not really a false assumption that the Earth was at the center, but a way of defining an Earth-centered coordinate system that is a completely legitimate way of recording observations.

The motion of the Earth was one of the great scientific issues in the history of mankind, but it is nearly always misrepresented.

This Babylon Bee parody is a lot more entertaining on the subject. To understand it, it helps to have seen the November 3, 1961 episode of The Twilight Zone.

Israeli prize nominee is quantum skeptic

Scott Aaronson write:

Oded Goldreich is a theoretical computer scientist at the Weizmann Institute in Rehovot, Israel. He’s best known for helping to lay the rigorous foundations of cryptography in the 1980s, ... Since then, I’ve interacted with Oded from time to time, partly around his firm belief that quantum computing is impossible.

Last month a committee in Israel voted to award Goldreich the Israel Prize (roughly analogous to the US National Medal of Science), for which I’d say Goldreich had been a plausible candidate for decades. But alas, Yoav Gallant, Netanyahu’s Education Minister, then rather non-gallantly blocked the award, solely because he objected to Goldreich’s far-left political views (and apparently because of various statements Goldreich signed, including in support of a boycott of Ariel University, which is in the West Bank). ...

[Nick] Is there any kind of correlation between leftist political views and QC skepticism?

Nick #33: I can’t say I’ve noticed any such correlation. On the other hand, maybe not surprisingly, I have noticed a strong correlation between QC skepticism and just general contrarianism, about politics, climate science, high-energy physics, or whatever else.

Some people just don't go along with the program for what everyone is supposed to believe, I guess.

Most of Aaronson's post and comments have to do with whether professors should be denied academic prizes because of their political opinions. This is how far we have gone. No bright young ambitious academic researcher expresses a politically incorrect opinion anymore.

Philosophers try to discredit Realism

John Horgan writes in SciAm:

Although my realism has been wobbling lately, I remain a realist. ...

Filmmaker Errol Morris, who studied under Kuhn in the 1970s and ended up loathing him, contends that Kuhnian-style postmodernism makes it easier for politicians and other powerful figures to lie. Philosopher Timothy Williamson makes a similar point in “In defence of realism.” “Imagine a future,” Williamson writes, “where a dictator or would-be dictator, accused of spreading falsehoods, can reply: ‘You are relying on obsolescent realist ideas of truth and falsity; realism has been discredited in philosophy.’”

I agree with methat, but I am afraid it is a losing battle.

Not only are philosohers denying realism, so are physicists, increasingly. And even those who agree with me on interpretations of quantum mechanics have conceded the term realism. That is, they will say that Copenhagen is not a realist interpretation, because we cannot simultaneously say the electron's position and momentum are.

Good videos about Quantum Mechanics

I have criticized popular accounts of quantum mechanics, but they are not all bad.

Lubos Motl praises a series of 3 elementary videos.

I also recommend Quantum Mechanics Isn’t Weird, We’re Just Too Big | Qiskit Seminar Series with Phillip Ball. Ball is a well-known science writer.

I am sure that there are many others. There have been good textbooks since 1930. Just be wary of anything talking about cats, parallel universes, and nonlocality.

There are lots of good videos on relativity, but I have a quibble with this one on general relativity mishaps. Most of it is about distinguishing the time dilation from velocity, which it calls special relativity, from the time dilations from gravity, which it calls general relativity.

He says that if the GPS satellites were the right height, the the effects would cancel out.

All that is correct, except that both time dilations are part of what used to be called special relativity. You don't need any metric geometry, and Einstein derived the gravity time dilation from just special relativity.

Some people say that special relativity is just about constant velocities (ie, uniform motion), but it was applied to accelating objects from the very start. The GPS satellite is just an accelating object. So is the ground receiver, if you figure in the acceleration of gravity.

Civilization was inspired by Astronomy

Thought experiment:

Imagine that over the last 11 thousand years (that is, the period of stable climate following upon the last ice age which allowed the human civilisation to develop)the atmospheric conditions on Earth were different: the skies were always covered, even in the absence of clouds, by a very light haze, not preventing the developmentof agriculture, but obscuring the stars and turning the sun and the moon into amorphous light spots. Would mathematics have had a chance to develop beyond basic arithmetic and geometry sufficient for measuring fields and keeping records of harvest? I doubt that. Civilisations which developed serious mathematics also had serious astronomy (it was an equivalent of our theoretical physics). But I claim even more: the movement of stars in the sky was the paradigm of precision andreproducibility, the two characteristic features of mathematics. Where else could humans learn the concept of absolute precision?

I agree with that. Without astronomy, we would not have much civilization today.

Trusting CDC to be science-based

From a Time magazine article, a year ago:

As the new coronavirus COVID-19 spreads in the U.S., people who are well want to stay that way. But since no vaccines are currently available, the strongest weapons Americans have are basic preventive measures like hand-washing and sanitizing surfaces, according to the Centers for Disease Control and Prevention (CDC).

The simplicity of those recommendations is likely unsettling to people anxious to do more to protect themselves, so it’s no surprise that face masks are in short supply—despite the CDC specifically not recommending them for healthy people trying to protect against COVID-19. “It seems kind of intuitively obvious that if you put something — whether it’s a scarf or a mask — in front of your nose and mouth, that will filter out some of these viruses that are floating around out there,” says Dr. William Schaffner, professor of medicine in the division of infectious diseases at Vanderbilt University. The only problem: that’s not effective against respiratory illnesses like the flu and COVID-19. If it were, “the CDC would have recommended it years ago,” he says. “It doesn’t, because it makes science-based recommendations.”

The science, according to the CDC, says that surgical masks won’t stop the wearer from inhaling small airborne particles, which can cause infection.

Remember this, next time you are told to trust the experts at the CDC.

Maybe the CDC was right that the masks were useless. The evidence for masks is dubious. I am not sure myself. But the official CDC recommendations don't seem to be any better than common-sense judgments from the average person.

Hard Science Journals have gone Leftist

Respected publications that used to stick to hard science are increasingly politicized.

The American Institute of Physics (AIP) published:

Gun violence rises in TV dramas over two decades, paralleling US gun homicide trends

. Historian Clayton Cramer points out that US gun homicides did not increase over the same period.

Scientific American currently has articles on Human Genetics Needs an Antiracism Plan and Trans Girls Belong on Girls’ Sports Teams. These articles advocate extreme leftist political positions that even most leftists would have rejected several years ago.

I would not mind this so much if the journals would occasionally publish right-wing views. But they do not. It is all Leftism, all the time.

Cashing in on the Quantum Hype

A quantum computing company is going public:

IonQ has entered into a definitive merger agreement with dMY Technology Group III (NYSE: DMYI.U)
IonQ To Become The First Publicly Traded Pure-Play Quantum Computing Company

Scott Aaronson is getting concerned about the unbound money and enthusiasm compromising the legitimate science:

What’s new is that millions of dollars are now potentially available to quantum computing researchers, along with equity, stock options, and whatever else causes “ka-ching” sound effects and bulging eyes with dollar signs. And in many cases, to have a shot at such riches, all an expert needs to do is profess optimism that quantum computing will have revolutionary, world-changing applications and have them soon. ...

As some of you might’ve seen already, IonQ, the trapped-ion QC startup that originated from the University of Maryland, is poised to have the first-ever quantum computing IPO — a so-called “SPAC IPO,” which while I’m a financial ignoramus, apparently involves merging with a shell company and thereby bypassing the SEC’s normal IPO rules. Supposedly they’re seeking $650 million in new funding and a $2 billion market cap. If you want to see what IonQ is saying about QC to prospective investors, click here. Lacking any choice in the matter, I’ll probably say more about these developments in a future post.

Meanwhile, PsiQuantum, the Palo-Alto-based optical QC startup, has said that it’s soon going to leave “stealth mode.” And Amazon, Microsoft, Google, IBM, Honeywell, and other big players continue making large investments in QC—treating it, at least rhetorically, not at all like blue-sky basic research, but like a central part of their future business plans.

I think that this is obviously a bubble, as none of these efforts have any reasonable chance of making money in the foreseeable future. But when it might burst, I cannot say.

Tesla Motors and GameStop also seem like bubbles. But they at least sell products successfully.

In June 2019, quantum computing progress was said to be doubly exponential. That is, the exponent is another exponent. If that were true, then these companies would be cashing in already.

Aaronson hints that he might be selling out himself. I would say that he is a fool if he does not cash in somehow. These bubbles/opportunities do not come often.

Update: The Aaronson blog generated these comments:

[Tamas V] I teach intro-level courses on QC, and once a participant mentioned that quantum computers could be used by fortune tellers. You can create a nice story, e.g. “nobody in the universe knows what the result of measuring a |+⟩ state in the computational basis will be… so it must be God’s way of giving you advice when I run this and that quantum program, because God is beyond our universe…”. And this can be done even with today’s devices, but not with pseudo- or classical randomness! Now it’s your part to convince the believers that God has nothing to do with the results

This example made me think that there might be some early unexpected commercial applications of QC, it’s only a matter of imagination…

[Scott] The problem with the fortune-telling “application” is not what you think it is. It’s that there a quantum random-number generator (let’s say, a Geiger counter next to a chunk of uranium) is all you need, and that’s been available for more than a century. A quantum computer simply isn’t needed for this.

[Tamas V] OMG, don’t approach it from the scientific side, it’s all about marketing.

1. You cannot *sell* such an idea with Geiger counters. Quantum computers are absolutely necessary, and they also allow the fortune teller to program more complicated/obscure “predictions”.
2. One cannot easily access and program a bunch of radioactive atoms via the cloud.

That could be a metaphor for the whole field. You cannot find anything useful, or sell anything honestly. The whole trick is to present something useless as magical.

Defining Mathematics

New Yorker magazine essay tries to define mathematics:

Mathematicians know what mathematics is but have difficulty saying it. I have heard: Mathematics is the craft of creating new knowledge from old, using deductive logic and abstraction. The theory of formal patterns. Mathematics is the study of quantity. A discipline that includes the natural numbers and plane and solid geometry. The science that draws necessary conclusions. Symbolic logic. The study of structures. The account we give of the timeless architecture of the cosmos. The poetry of logical ideas. Statements related by very strict rules of deduction. A means of seeking a deductive pathway from a set of axioms to a set of propositions or their denials. A science involving things you can’t see, whose presence is confined to the imagination. A proto-text whose existence is only postulated. A precise conceptual apparatus. The study of ideas that can be handled as if they were real things. The manipulation of the meaningless symbols of a first-order language according to explicit, syntactical rules. A field in which the properties and interactions of idealized objects are examined. The science of skillful operations with concepts and rules invented for the purpose. Conjectures, questions, intelligent guesses, and heuristic arguments about what is probably true. The longest continuous human thought. Laboriously constructed intuition. The thing that scientific ideas, as they grow toward perfection, become. An ideal reality. A story that has been written for thousands of years, is always being added to, and might never be finished. The largest coherent artifact that’s been built by civilization. Only a formal game. What mathematicians do, the way musicians do music.

Those are good, but this gets to the heart of the matter:

Mathematicians live within a world that is essentially certain. The rest of us, even other scientists, live within one where what represents certainty is so-far-as-we-can-tell-this-result-occurs-almost-all-of-the-time. Because of mathematics’ insistence on proof, it can tell us, within the range of what it knows, what happens time after time.

I will have to ponder this:

In Book 7 of the Republic, Plato has Socrates say that mathematicians are people who dream that they are awake.

Denyse O'Leary writes:

The Oregon Department of Education (ODE) recently encouraged teachers to register for training that encourages “ethnomathematics,” an education trend that argues, “among other things, that White supremacy manifests itself in the focus on finding the right answer”:

“The concept of mathematics being purely objective is unequivocally false, and teaching it is even much less so,” the document for the “Equitable Math” toolkit reads. “Upholding the idea that there are always right and wrong answers perpetuate objectivity as well as fear of open conflict.” …

An associated “Dismantling Racism” workbook, linked within the toolkit, similarly identifies “objectivity” — described as “the belief that there is such a thing as being objective or ‘neutral'” — as a characteristic of White supremacy.

Instead of focusing on one right answer, the toolkit encourages teachers to “come up with at least two answers that might solve this problem.”

Sam Dorman, “Oregon promotes teacher program that seeks to undo ‘racism in mathematics’” at Fox News (February 11, 2021)

Opponents of the new trend offer varying accounts of its origin — perhaps it results from in changes in overall philosophy of life or perhaps from the practical need to placate teachers’ unions, which may have various objectives apart from enabling numeracy in students. No matter, not only are x and y under attack but so is 2 + 2 = 4.

It is embarrassing how many respected scientists allow this nonsense to persist without saying anything. I am happy to see that physicist Lawrence Krauss wrote a Quillette essay denouncing some of the leftist propaganda:

Social justice activists have been arguing for some time that scientific societies and institutions need to address systemic sexism and racism in STEM disciplines. However, their rationale is often anything but scientific. For example, whenever percentages in faculty positions, test scores, or grant recipients in various disciplines do not match percentages of national average populations, racism or sexism is generally said to be the cause. This is in spite of the fact that no explicit examples of racism or sexism generally accompany the statistics. Correlation, after all, is not causation. Without some underlying mechanism or independent evidence to explain a correlation of observed outcomes with population statistics, inferring racism or sexism in academia as the cause is inappropriate.

One might have hoped for more rigor from the leadership of scientific societies and research institutions. Alas, this has not been the case.

My guess is that other scientists are afraid to say this out of fear that they will be canceled. Krauss only dares because he has already been canceled. His enemies tried to get him fired, based on gossip. I don't know if they succeeded or not.