A quick comment on the tweet about number theory papers, which in fact noticeably understates the scope of the situation.

-- Actually 100 pages is nowhere near the upper limit.

-- Actually I *wish* I could "deeply understand" some of these papers (even ones in my own subfield!) merely by taking a semester-long course.

-- It's most of math, not just number theory, and it's not just because the papers are long. The theory required to even *state* the results is piled so deep that one cannot reasonably expect to check all of it for oneself.

-- Nevertheless, I'm surprisingly confident that most of it is basically true, even though the literature is probably filled with superficial errors. This is because it's all interconnected and everyone spot-tests bits and pieces of the edifice as they poke around in their tiny domain. A truly catastrophic problem would eventually lead to a contradiction and thus be noticed.

-- While there technically exist the *very, very rare* occasional mathematician who resists admitting error, and subfields where errors are rampant and confidence is low, the extent of uncertainty still operates at many orders of magnitude below what is normal in any other area of science; there is simply no plausible comparison. For this I am very grateful.

Two good examples in mathematics are the proof of Fermat's famous Last Theorem and the proof of the ABC Conjecture. They are both huge proofs and were originally presented at multi-day seminars, but Fermat's theorem is generally accepted as proven while the proof of the ABC Conjecture is considered "problematic".

The proof of Fermat's theorem came from an expected corner involving elliptic curves and modular forms. It's general structure was proposed decades before the accepted proof. There were lots of mathematicians familiar with and working in the area. When the proof was first presented, a flaw was discovered but apparently patched to general satisfaction in the field.

The proof of the ABC Conjecture came from a single mathematician largely working alone in an area that he developed and expanded. While there are mathematicians working in the same general area as this lone researcher, decades of isolated research mean that validating the proof requires a lot of catching up. There was a lot of initial excitement, seminars, critiques and so on, but, when I last checked, the general feeling was that it is possible that the ABC Conjecture has been proven, most mathematicians regard it is as yet unproven.

This is how the scientific endeavor should work. It's both an individual and a joint effort. People come up with ideas, develop and present them. Then, the community collectively decides whether to accept the results, reject it or consider the matter as yet open. The proof of Fermat's theorem was quickly accepted. The proof of the ABC Conjecture is still considered open, though there seems to be an interesting new approach that is worth studying and may yet prove it.

Do you see theorem provers (Lean, etc.) catching on in your field anytime soon?

As I understand it, right now they’re basically of interest to people who research theorem provers. But there’s that group at Imperial College trying to make them usable for working mathematicians…

Number theory is devoted to studying the nature or structure of "numbers" as a concept, often specifically the set integers, although sometimes it blends over into the more abstract set of things that behaving kind-of like the set of integers.

As for what it's used for -- probably the most obvious practical application is the entire field of public-key cryptography, which relies on the fact that factoring the product of two large prime numbers is hard (and number theory tells us it _should_ be hard, absent a quantum computer with enough qubits that it can, in effect, test all possible factors at once).

On the more abstract side, you can get into first Group theory:

You have a set of elements, and an operation, such that the "product" of that operation on any two elements, produces another element of the group, and exactly one element is the "identity", where a*i = i*a = a.

And then that can be further developed into the theory of Fields, which have _two_ operations, which are defined to have some properties similar to addition and multiplication. https://en.wikipedia.org/wiki/Field_(mathematics)

Groups and Fields get interesting though when you find some examples that work _kind_ of like the integer numbers, but not quite. So, for instance, let's define a group that represents "ways you could orient a triangle".

So, to begin with, we will define the identity element as "1, 2, 3". We'll label corner 1 at the top, and then go around clockwise with corners 2 and 3.

We then define element a to represent a clockwise rotation. So a = a*i = i*a = "3, 1, 2". Corner 3 has rotated around to the top.

Element b represents a flip around the vertical access. b = b*i = i*b = "1, 3, 2". Corner 1 is still at the top, but corners 2 and 3 have traded places.

Now, ask: What is a*b? Well, first we rotate to "3, 1, 2", and then we flip, ending up at "3, 2, 1".

But what is b*a? Well, first we flip to "1, 3, 2", and then we rotate to "2, 1, 3".

Aha! Those are not the same thing! This property, that a*b is NOT always the same as b*a, is called "non-abelian". (By contrast, a group where a*b=b*a in all cases is "abelian".) And of course this is quite different from how integer arithmetic behaves.

However, note that we have been talking about rotating and flipping an object -- you could cut out a paper triangle and label the corners, to show how this works.

This type of non-abelian group has major applications in machine tooling. If you ever see one of those automated cutting / lathing type machines that can pick up an object and manipulate it in space, it is relying heavily on a set of mathematical constructs that build off the insight I've described above. Orientations of a triangle is literally the simplest possible example of this type of group. :-)

Personally I think a pair of Ray-Ban’s would suit Noah better than the blue reflectors when it comes to sunglasses.

And let me join into the self congratulations I also knew that the mask thing wasn’t true. You would not believe how many Facebook arguments I got into people about that.

I think we’re scientist get in trouble is when they conflate science with social policy. My big one would be mask mandates. I’m vaccinated, and I hate masks. Now I have dug into all of the numbers, and I know that if you vaccinated do you have way less chance of getting it and spreading Covid then you would as a non-vaccinated person wearing a mask. Therefore I feel quite comfortable not wearing masks, and not guilty at all for not wearing them.

But inevitably someone will say science says you can still get Covid after you’ve been vaccinated.

Oh anyway, I digress. Vaccinations has just become this weird partisan issue.

So right now I’m working with three other Americans here in Brazil. So I took a survey and case study.

First a general observation. None of us know anyone that has died of COVID. We all know in passing one or two person who was hospitalized and released. Think coworker you see occasionally.

Me. 51. White dude. Swing voter. Trump disliker. But not liberal. Vaccinated. Did a lot of research. Never had COVID.

Engineer: 27. White Democrat. Standard. Not super liberal, but mainstream. Vaccinated. Never had COVID.

Winder 1. 35. Black. Moderate to conservative. Had COVID. Not vaccinated. Fairly knowledgeable. He figures they will tweak the vaccine in future to handle all the variants and once it’s approved will probably get it then. Knows he is somewhat protected. Comfortable with risk.

Winder 2. 35. White. Conservative. Likes Trump. Hates Biden. Had COVID. Not vaccinated. Knows he is somewhat protected. Waiting for FDA approval.

None of them had any conspiracy theories about the vaccinations. None are overly political. Conversations are about chasing Brazilians women instead of Politics. All were basically familiar with the issues. No science deniers.

Quite frankly when I spoke to them, each person seemed pretty rational to me. It was more about risk vs unknown risk.

Anyway, what the rabid vaccinator and mask preachers miss and don’t really comprehend is that even though there is a pandemic…. For the majority of people they don’t see it or experience it. Their love ones aren’t dying. Even if someone’s knows someone who died of COVID it was someone who was very old or sick already. I don’t think policy makers really grasp this. How so much of the doom and gloom comes across as overblown. This is how conservatives can rationalize not getting vaccinated.

Anyway, I am typing this on my phone, so forgive any spelling and grammar mistakes.

Winder 1 just asked for a copy of the email. May update after he reads it.

Also. There is a tweet going around talking about people having cross partisan friendships. I definitely don’t have to worry about that. I love my job and the people I work with. Long hours. Away from home. But working in Energy contributes to society. Diverse coworkers.

Your analysis is good, but it's all directed at whether to doubt the experts. At least equally important is whether you have grounds to trust yourself, your own judgment over that of experts. IMHO it's likely if you disagree with the experts, particularly if you get emotionally involved, your chances of being wrong get high.

It seems to me that the main problem with trust in science right now is that red Americans correctly perceive that most scientists are the sort of blue Americans who hold in their heart lots of partisan hatred toward red Americans. E.g. they would be livid if their child dated a red person, they smile when they read a news article about red people acting stupidly or suffering misfortune, and (most importantly for present purposes) they would be thrilled if one of their discoveries implied that red Americans are misguided.

I suspect this is the main reason why red Americans don't trust new, clearly proven scientific results in the current environment. I don't think it's the explanation for how red distrust of science began--that had a lot to do with religion and opposition to environmentalism--but it's what sustains that distrust today.

I had written a similar topic on my Medium of the problem of liberals turning science into a partial religion. "Trust the science. Believe in science." It does seem to turn objectively good things like vaccines into something that should be taken just on faith rather than data. Worse is that it primes people to distrust the scientific community in other avenues like climate if a consensus opinion is ever not correct.

One note, though, on the example of trusting the experts on vaccines: the expert scientists who designed the vaccines, took the data, and published the papers were very trustworthy! But the "expert" heath communicators of various stripes spent months undermining the vaccines by downplaying how effective they were, and the degree to which being vaccinated allowed for an almost entirely normal life.

On your list of four factors, it may be worth pointing out that #2 is clearly and by far the most important one!

The pandemic provides a good example to illustrate this: anyone who strongly doubted the effectiveness of the vaccines was obviously foolish, but one could've made great money betting against SIR epidemiological models when it came to predicting the course of the pandemic. The SIR model scores great on criteria 1, 3 and 4, but it turns out to have terrible empirical support (anyone could look back at the 2009 H1N1 pandemic curve and see that it was nothing like what the models would predict for the measured R0 value).

Great post. Especially liked the last paragraph. I think skepticism is trained into scientists. One has to be reasonably comfortable with holding two contradictory thoughts in one’s mind simultaneously; that scientific explanation X seems to be the best one we have, and... it might be wrong. Maybe doesn’t offer us certainty, but this method is the best one we have and the use of it has led to almost unbelievable progress in understanding the world. I personally like reading and thinking about economics, but there are so few well-established, widely accepted explanations that have good predictive power about how economies work and will work that I’m worried it doesn’t qualify as a science.

Great article! I'm a retired English professor curious about economics and science who, of course, lacks the background, especially the math, to understand much. I approach the problem from a slightly different direction but with much the same result: I look at the rhetoric of whatever I'm reading. Who's the author and what do I know about them? Is the argument coherent? Is it logical? These last two are very important. A coherent argument will frame the question carefully and concede what isn't known or where error might lie. As for logic, it's amazing how many questions go begged and how many ad hominem fallacies and false dilemmas lead to unwarranted conclusions. By these standards, articles in Scientific American and Krugman's op eds are especially persuasive. Sadly, because so much of conservative rhetoric has followed Limbaugh, most writing on the right is trash, including some by economists with sterling credentials.

A quick comment on the tweet about number theory papers, which in fact noticeably understates the scope of the situation.

-- Actually 100 pages is nowhere near the upper limit.

-- Actually I *wish* I could "deeply understand" some of these papers (even ones in my own subfield!) merely by taking a semester-long course.

-- It's most of math, not just number theory, and it's not just because the papers are long. The theory required to even *state* the results is piled so deep that one cannot reasonably expect to check all of it for oneself.

-- Nevertheless, I'm surprisingly confident that most of it is basically true, even though the literature is probably filled with superficial errors. This is because it's all interconnected and everyone spot-tests bits and pieces of the edifice as they poke around in their tiny domain. A truly catastrophic problem would eventually lead to a contradiction and thus be noticed.

-- While there technically exist the *very, very rare* occasional mathematician who resists admitting error, and subfields where errors are rampant and confidence is low, the extent of uncertainty still operates at many orders of magnitude below what is normal in any other area of science; there is simply no plausible comparison. For this I am very grateful.

Two good examples in mathematics are the proof of Fermat's famous Last Theorem and the proof of the ABC Conjecture. They are both huge proofs and were originally presented at multi-day seminars, but Fermat's theorem is generally accepted as proven while the proof of the ABC Conjecture is considered "problematic".

The proof of Fermat's theorem came from an expected corner involving elliptic curves and modular forms. It's general structure was proposed decades before the accepted proof. There were lots of mathematicians familiar with and working in the area. When the proof was first presented, a flaw was discovered but apparently patched to general satisfaction in the field.

The proof of the ABC Conjecture came from a single mathematician largely working alone in an area that he developed and expanded. While there are mathematicians working in the same general area as this lone researcher, decades of isolated research mean that validating the proof requires a lot of catching up. There was a lot of initial excitement, seminars, critiques and so on, but, when I last checked, the general feeling was that it is possible that the ABC Conjecture has been proven, most mathematicians regard it is as yet unproven.

This is how the scientific endeavor should work. It's both an individual and a joint effort. People come up with ideas, develop and present them. Then, the community collectively decides whether to accept the results, reject it or consider the matter as yet open. The proof of Fermat's theorem was quickly accepted. The proof of the ABC Conjecture is still considered open, though there seems to be an interesting new approach that is worth studying and may yet prove it.

Do you see theorem provers (Lean, etc.) catching on in your field anytime soon?

As I understand it, right now they’re basically of interest to people who research theorem provers. But there’s that group at Imperial College trying to make them usable for working mathematicians…

Two questions. What is number theory. And what is it used for?

Number theory is devoted to studying the nature or structure of "numbers" as a concept, often specifically the set integers, although sometimes it blends over into the more abstract set of things that behaving kind-of like the set of integers.

As for what it's used for -- probably the most obvious practical application is the entire field of public-key cryptography, which relies on the fact that factoring the product of two large prime numbers is hard (and number theory tells us it _should_ be hard, absent a quantum computer with enough qubits that it can, in effect, test all possible factors at once).

On the more abstract side, you can get into first Group theory:

You have a set of elements, and an operation, such that the "product" of that operation on any two elements, produces another element of the group, and exactly one element is the "identity", where a*i = i*a = a.

And then that can be further developed into the theory of Fields, which have _two_ operations, which are defined to have some properties similar to addition and multiplication. https://en.wikipedia.org/wiki/Field_(mathematics)

Groups and Fields get interesting though when you find some examples that work _kind_ of like the integer numbers, but not quite. So, for instance, let's define a group that represents "ways you could orient a triangle".

So, to begin with, we will define the identity element as "1, 2, 3". We'll label corner 1 at the top, and then go around clockwise with corners 2 and 3.

We then define element a to represent a clockwise rotation. So a = a*i = i*a = "3, 1, 2". Corner 3 has rotated around to the top.

Element b represents a flip around the vertical access. b = b*i = i*b = "1, 3, 2". Corner 1 is still at the top, but corners 2 and 3 have traded places.

Now, ask: What is a*b? Well, first we rotate to "3, 1, 2", and then we flip, ending up at "3, 2, 1".

But what is b*a? Well, first we flip to "1, 3, 2", and then we rotate to "2, 1, 3".

Aha! Those are not the same thing! This property, that a*b is NOT always the same as b*a, is called "non-abelian". (By contrast, a group where a*b=b*a in all cases is "abelian".) And of course this is quite different from how integer arithmetic behaves.

However, note that we have been talking about rotating and flipping an object -- you could cut out a paper triangle and label the corners, to show how this works.

This type of non-abelian group has major applications in machine tooling. If you ever see one of those automated cutting / lathing type machines that can pick up an object and manipulate it in space, it is relying heavily on a set of mathematical constructs that build off the insight I've described above. Orientations of a triangle is literally the simplest possible example of this type of group. :-)

(This also leads into my favorite math joke: What's purple, commutes, and has exactly five devoted followers? A finitely venerated abelian grape.)

Personally I think a pair of Ray-Ban’s would suit Noah better than the blue reflectors when it comes to sunglasses.

And let me join into the self congratulations I also knew that the mask thing wasn’t true. You would not believe how many Facebook arguments I got into people about that.

I think we’re scientist get in trouble is when they conflate science with social policy. My big one would be mask mandates. I’m vaccinated, and I hate masks. Now I have dug into all of the numbers, and I know that if you vaccinated do you have way less chance of getting it and spreading Covid then you would as a non-vaccinated person wearing a mask. Therefore I feel quite comfortable not wearing masks, and not guilty at all for not wearing them.

But inevitably someone will say science says you can still get Covid after you’ve been vaccinated.

Oh anyway, I digress. Vaccinations has just become this weird partisan issue.

So right now I’m working with three other Americans here in Brazil. So I took a survey and case study.

First a general observation. None of us know anyone that has died of COVID. We all know in passing one or two person who was hospitalized and released. Think coworker you see occasionally.

Me. 51. White dude. Swing voter. Trump disliker. But not liberal. Vaccinated. Did a lot of research. Never had COVID.

Engineer: 27. White Democrat. Standard. Not super liberal, but mainstream. Vaccinated. Never had COVID.

Winder 1. 35. Black. Moderate to conservative. Had COVID. Not vaccinated. Fairly knowledgeable. He figures they will tweak the vaccine in future to handle all the variants and once it’s approved will probably get it then. Knows he is somewhat protected. Comfortable with risk.

Winder 2. 35. White. Conservative. Likes Trump. Hates Biden. Had COVID. Not vaccinated. Knows he is somewhat protected. Waiting for FDA approval.

None of them had any conspiracy theories about the vaccinations. None are overly political. Conversations are about chasing Brazilians women instead of Politics. All were basically familiar with the issues. No science deniers.

Quite frankly when I spoke to them, each person seemed pretty rational to me. It was more about risk vs unknown risk.

Anyway, what the rabid vaccinator and mask preachers miss and don’t really comprehend is that even though there is a pandemic…. For the majority of people they don’t see it or experience it. Their love ones aren’t dying. Even if someone’s knows someone who died of COVID it was someone who was very old or sick already. I don’t think policy makers really grasp this. How so much of the doom and gloom comes across as overblown. This is how conservatives can rationalize not getting vaccinated.

Anyway, I am typing this on my phone, so forgive any spelling and grammar mistakes.

Winder 1 just asked for a copy of the email. May update after he reads it.

Also. There is a tweet going around talking about people having cross partisan friendships. I definitely don’t have to worry about that. I love my job and the people I work with. Long hours. Away from home. But working in Energy contributes to society. Diverse coworkers.

Your analysis is good, but it's all directed at whether to doubt the experts. At least equally important is whether you have grounds to trust yourself, your own judgment over that of experts. IMHO it's likely if you disagree with the experts, particularly if you get emotionally involved, your chances of being wrong get high.

It seems to me that the main problem with trust in science right now is that red Americans correctly perceive that most scientists are the sort of blue Americans who hold in their heart lots of partisan hatred toward red Americans. E.g. they would be livid if their child dated a red person, they smile when they read a news article about red people acting stupidly or suffering misfortune, and (most importantly for present purposes) they would be thrilled if one of their discoveries implied that red Americans are misguided.

I suspect this is the main reason why red Americans don't trust new, clearly proven scientific results in the current environment. I don't think it's the explanation for how red distrust of science began--that had a lot to do with religion and opposition to environmentalism--but it's what sustains that distrust today.

I had written a similar topic on my Medium of the problem of liberals turning science into a partial religion. "Trust the science. Believe in science." It does seem to turn objectively good things like vaccines into something that should be taken just on faith rather than data. Worse is that it primes people to distrust the scientific community in other avenues like climate if a consensus opinion is ever not correct.

One note, though, on the example of trusting the experts on vaccines: the expert scientists who designed the vaccines, took the data, and published the papers were very trustworthy! But the "expert" heath communicators of various stripes spent months undermining the vaccines by downplaying how effective they were, and the degree to which being vaccinated allowed for an almost entirely normal life.

On your list of four factors, it may be worth pointing out that #2 is clearly and by far the most important one!

The pandemic provides a good example to illustrate this: anyone who strongly doubted the effectiveness of the vaccines was obviously foolish, but one could've made great money betting against SIR epidemiological models when it came to predicting the course of the pandemic. The SIR model scores great on criteria 1, 3 and 4, but it turns out to have terrible empirical support (anyone could look back at the 2009 H1N1 pandemic curve and see that it was nothing like what the models would predict for the measured R0 value).

Great post. Especially liked the last paragraph. I think skepticism is trained into scientists. One has to be reasonably comfortable with holding two contradictory thoughts in one’s mind simultaneously; that scientific explanation X seems to be the best one we have, and... it might be wrong. Maybe doesn’t offer us certainty, but this method is the best one we have and the use of it has led to almost unbelievable progress in understanding the world. I personally like reading and thinking about economics, but there are so few well-established, widely accepted explanations that have good predictive power about how economies work and will work that I’m worried it doesn’t qualify as a science.

Great article! I'm a retired English professor curious about economics and science who, of course, lacks the background, especially the math, to understand much. I approach the problem from a slightly different direction but with much the same result: I look at the rhetoric of whatever I'm reading. Who's the author and what do I know about them? Is the argument coherent? Is it logical? These last two are very important. A coherent argument will frame the question carefully and concede what isn't known or where error might lie. As for logic, it's amazing how many questions go begged and how many ad hominem fallacies and false dilemmas lead to unwarranted conclusions. By these standards, articles in Scientific American and Krugman's op eds are especially persuasive. Sadly, because so much of conservative rhetoric has followed Limbaugh, most writing on the right is trash, including some by economists with sterling credentials.

"or whether it’s still mostly in the theoretical stage (like macroeconomics)"

I resent that