I Know the Math Says so, but Is It Really True?

[PeterDonis]

I’m sure anyone who has hung out long enough here on Physics Forums has encountered threads that go something like this (I’ll use an example based on threads I’ve seen and participated in in the relativity forum, but I’m sure similar things occur in other forums as well):
Original Poster: I don’t understand how black holes can exist. Doesn’t it take an infinite time for anything to fall in?
SA/Mentor: The “infinite time” is just coordinate time; if you calculate the proper time experienced by the infalling object when it reaches the horizon, it comes out finite.
[Exchange follows in which the actual math may even be shown or linked to.]
Original Poster: Sorry, I don’t understand all that math. Can’t you explain it in plain language? If you can’t put it in terms that make sense to me, I don’t believe it, no matter what your math says.
(Please note, the above are not direct quotes, and I am not going to name any names because I have no desire to single anybody...

Continue reading...

 

[phinds]

Excellent article Peter and one I'm sure we'll get a lot of use out of.

Personally, I think you should be stronger in the final sentence. "Sorry, but that’s the way it is." just sounds weak to me. I'd say something more like what I think Feynman would say: "You may not like it, but that's the way it is".

 

[PeterDonis]

I'd say something more like what I think Feynman would say: "You may not like it, but that's the way it is".

I'm not sure "sorry" was "weaker", exactly, but I agree this wording is better, so I've edited the article accordingly.

 

[mpresic3]

Now that COVID has hit, it has allowed me to study General Relativity, a subject I have never examined in detail before. I am using Sean Carroll's textbook, Spacetime and Geometry, and backing it up with Wald's textbook that Carroll's book borrows heavily from.
Sometimes I have to read the same chapter and sub-chapters several times. I am struggling, and now I see the point of a saying I heard once, " I have to believe 5 improbable things before breakfast". I can understand now that sometimes our common sense misleads us.
Sometimes there are no easy explanations. I never really believed the line, "you don't really understand it until you can explain it to your grandmother". BTW, grandmothers are not necessarily dense. Madame Curie, Emmy Noether, and Maria Goeppert Mayer were probably (I would have to look it up), someone's grandmother. I know several grandmom's that beat their grandkids at chess.

 

[eltodesukane]

"Original Poster: I don’t understand how black holes can actually exist. Doesn’t it take an infinite time for anything to fall in?
SA/Mentor: The “infinite time” is just coordinate time; if you calculate the proper time experienced by the infalling object when it reaches the horizon, it comes out finite."

-- But still, from our own coordinate frame, no back hole has yet formed, and it will still be so in a trillion trillion trillion trillion years. Yet I read that black holes will start evaporating by then. How is that possible in our own proper time?

 

[phinds]

-- But still, from our own coordinate frame, no back hole has yet formed, and it will still be so in a trillion trillion trillion trillion years. Yet I read that black holes will start evaporating by then. How is that possible in our own proper time?

Just as an FYI aside "a trillion trillion trillion trillion years" is basically zero (not even a rounding error) compared to the amount of time it takes for Hawking Radiation to become significant.

 

[PeterDonis]

But still, from our own coordinate frame, no back hole has yet formed, and it will still be so in a trillion trillion trillion trillion years. Yet I read that black holes will start evaporating by then. How is that possible in our own proper time?

There are plenty of other PF threads already answering this question, not to mention this Insights article:

https://www.physicsforums.com/insights/black-holes-really-exist/

Further discussion of this question should occur either in the comment thread on the Insights article, or in a new thread. This discussion thread is not about the specific physics question I used for an illustration in the subject article, but about the general point the article is making.

 

[Jarvis323]

To play devils advocate, if we take the article completely seriously, ...

There are a lot of Physics theories. If one couldn't have an opinion about any of them without knowing all of the math, then there would be no reason for them to choose to learn one over another. People would be forced to just learn completely whatever random theory comes up (no matter how ridiculous), without using any reason at all to make the choice.

Obviously there is a world of middle ground and subtlety that goes between the lines.

 

[PeterDonis]

There are a lot of Physics theories. If one couldn't have an opinion about any of them without knowing all of the math, then there would be no reason for them to choose to learn one over another.

I don't see why. Knowing enough about the subject matter of a theory to know whether one is interested in it, is a lot easier than knowing enough about the details of the math to have a valid opinion, based on your own knowledge, about whether the theory is correct.

 

[Jarvis323]

I mean, technically correctness shouldn't be left to opinion at all. But the correctness of the math of the theory isn't the same as the validity of the theory, its promise relative to other competing theories, or its relationship to objective reality.

 

[phinds]

... the correctness of the math of the theory isn't the same as ... its relationship to objective reality.

I see your point, but I disagree with the thrust of your conclusion. If the math of the theory does not completely correspond (within experimental limits) to objective reality then the theory gets tossed out (or is superseded by a more refined theory, which is what happened when GR replaced Newtonian Gravity)

 

[Rive]

If one couldn't have an opinion about any of them without knowing all of the math, then there would be no reason for them to choose to learn one over another.

Well, sometimes when the competing approaches has similar area of validity and equivalent math, the reasoning will be based on natural laziness: which math looks easier to learn

Of course some people will still choose to continue with Hamilton when Newton falls short...

 

[PeterDonis]

correctness shouldn't be left to opinion at all

Who said it was?

Whether or not a particular mathematical model is "correct" in the sense of being self-consistent is something that can be objectively tested.

Whether or not a particular mathematical model is "correct" in the sense of making predictions that match reality is also something that can be objectively tested.

The only place opinions come in is if you have a mathematical model that can't be tested at present against reality because we don't have the technical capability to do the tests. (String theory comes to mind as an example.) Then people can have different opinions about how the tests might come out if and when we have the ability to do them.

But I wasn't talking about that case in the article. I was talking about the case where the mathematical model has been tested against reality, at least in some domain, and it has passed the tests--the theory has experimental confirmation--but the theory says things that are counterintuitive and the tests are not things that are part of people's everyday experience--or at least the link between the tests and people's everyday experience is not easy for people to grasp. (For example, GPS is now part of people's everyday experience, in the sense that people know their smartphones use GPS to accurately detect their location, but most people don't have an intuitive grasp of how that capability of GPS shows the correctness of General Relativity's predictions for spacetime geometry in the vicinity of the Earth.)

So when physicists talk about what this mathematical model predicts for cases that are, strictly speaking, outside the tested domain (we don't have direct experimental tests of GR's predictions at the horizon of a black hole), but are well within the expected domain of validity of the model (spacetime curvature at or near the horizon of a black hole of stellar mass or larger is many, many orders of magnitude smaller than the spacetime curvature at which GR's predictions are expected to break down), it's hard for ordinary people to understand that no, the physicists aren't just speculating, they are stating the unequivocal predictions of a model that has so far passed all the experimental tests we can throw at it, and yes, the model really does say what the physicists say it says, however counterintuitive it seems to a lay person.

 

[Ralph Dratman]

I don't see the problem with some individual not believing something in physics. Anyone is free to disbelieve. Why not?

I suppose one might worry that such an opinion could become widespread, something like vaccine rejection. Personally that does not worry me either, because disbelief in a scientific finding does not influence public health or anything like that.

 

[PeterDonis]

Anyone is free to disbelieve.

Certainly. But if they come here to PF and ask questions, and get answers, but are unwilling to accept the answers, that's something different. That's the kind of scenario I was talking about in the article.

disbelief in a scientific finding does not influence public health

Disbelief in something like the standard GR model of black holes probably doesn't influence public health, yes. But not all topics that are discussed here on PF as a whole (since PF includes subforums for topics other than theoretical physics) are that disconnected from practical matters like public health.

 

[russ_watters]

Perhaps a more direct response:

I suppose one might worry that such an opinion could become widespread, something like vaccine rejection. Personally that does not worry me either, because disbelief in a scientific finding does not influence public health or anything like that.

That is demonstrably false. The anti-vax position has had a measurable negative impact on public health in terms of reduced vaccination rates and increases in the incidences of the associated diseases.

 

[BWV]

There are disciplines outside of fundamental physics where math is used more as a metaphor or analogy- to simplify a complex process and highlight key relationships where the inputs may not be fully knowable - economics is an example in the social sciences, but you see this in biology as well - predator / prey models, modelling epidemics etc.

 

[phinds]

... disbelief in a scientific finding does not influence public health or anything like that.

You have SERIOUSLY not been paying attention to the news for the last decade.

 

[Fred Wright]

Perhaps a more direct response:

That is demonstrably false. The anti-vax position has had a measurable negative impact on public health in terms of reduced vaccination rates and increases in the incidences of the associated diseases.

I think the term "scientific finding" when connected with the term "public health" is ambiguous, to say the least. For example, about two months ago I read on a Russian news website that two independent groups from French universities surveyed European hospital records of Covid patients and discovered that only about 0.1% were smokers. They came to the conclusion that smoking cigarettes prevented infection of the virus and suggested that healthcare workers involved in the epidemic wear nicotine patches. This scientific finding is obviously contradictory to our accepted concept of public health.

Your assertion that the anti-vax position has a measurable negative impact on public health may be true or false. I don't have the data to make a judgement. I don't know the framework of how the data was acquired and I don't know the competence of the analysts. But to forbid doubt in the assertion raises it to the realm of religious belief.

 

[BWV]

Your assertion that the anti-vax position has a measurable negative impact on public health may be true or false. I don't have the data to make a judgement. I don't know the framework of how the data was acquired and I don't know the competence of the analysts. But to forbid doubt in the assertion raises it to the realm of religious belief.

You are welcome
https://ourworldindata.org/grapher/measles-cases-and-death-rate?yScale=log

 

[russ_watters]

But to forbid doubt in the assertion raises it to the realm of religious belief.

You are mischaracterizing the claim and therefore also my response. The claim was absolute: "does not", not merely an expression of doubt. And the example given just so happens to be extensively researched and unambiguous.

 

[PeterDonis]

There are disciplines outside of fundamental physics where math is used more as a metaphor or analogy- to simplify a complex process and highlight key relationships where the inputs may not be fully knowable - economics is an example in the social sciences, but you see this in biology as well - predator / prey models, modelling epidemics etc.

The mathematical models you refer to in other disciplines are still subject to the same test as mathematical models in physics: either they make predictions that match the data, or they don't. Models that don't make predictions at all aren't the kind of "math" I am talking about in the article.

Also, your post implies that mathematical models in physics don't have the characteristics you describe--simplifying complex processes, modeling domains where inputs are not fully knowable. That is quite wrong. There are plenty of domains in physics where the same issues arise. In fact, it's hard to find a domain even in physics where those issues don't arise.

 

[russ_watters]

Disbelief in something like the standard GR model of black holes probably doesn't influence public health, yes. But not all topics that are discussed here on PF as a whole (since PF includes subforums for topics other than theoretical physics) are that disconnected from practical matters like public health.

My broad perception and concern is that it is part of an overall philosophy of distrust combined with a limited ability to evaluate information. So it does not surprise me at all when I see examples where lack of "belief" in standard physics/science coincides with lack of belief/trust in science on public health matters or in other contexts. That's part of the reason I think basic science learning is so important; it teaches critical thinking skills that can be applied elsewhere.

 

[PeterDonis]

My broad perception and concern is that it is part of an overall philosophy of distrust combined with a limited ability to evaluate information.

Limited ability to evaluate information is a factor, I agree. I also think there are other factors that contribute to a philosophy of distrust in statements made by public authorities. One of those factors is that at least some public authorities have a track record of making statements which should not be taken at face value. Which makes it even more important for individual citizens to have critical thinking skills, so that they can evaluate individual statements from any source on the merits without having to rely on some kind of authoritative status of a source, since any such status can be misused if it allows statements made by that source to be taken as true without critical evaluation.

 

[BWV]

The mathematical models you refer to in other disciplines are still subject to the same test as mathematical models in physics: either they make predictions that match the data, or they don't. Models that don't make predictions at all aren't the kind of "math" I am talking about in the article.

Also, your post implies that mathematical models in physics don't have the characteristics you describe--simplifying complex processes, modeling domains where inputs are not fully knowable. That is quite wrong. There are plenty of domains in physics where the same issues arise. In fact, it's hard to find a domain even in physics where those issues don't arise.

I took your piece to be discussing theories that can be expressed mathematically - i.e. you cannot understand GR without understanding the math. but you can perfectly well understand what an epidemic is without understanding any of the various mathematical models of how they spread.

 

[PeterDonis]

you can perfectly well understand what an epidemic is without understanding any of the various mathematical models of how they spread

You can understand what an epidemic is, yes, but if someone has a particular mathematical model of how epidemics spread that makes predictions that have matched the data so far, and that model has some counterintuitive feature that makes you want to disbelieve its predictions about some possible future epidemic, you won't get very far criticizing the model if you don't understand the math.

 

[BWV]

You can understand what an epidemic is, yes, but if someone has a particular mathematical model of how epidemics spread that makes predictions that have matched the data so far, and that model has some counterintuitive feature that makes you want to disbelieve its predictions about some possible future epidemic, you won't get very far criticizing the model if you don't understand the math.

Yes, but the math is not the theory. The Germ Theory of Disease requires no math

 

[PeterDonis]

the math is not the theory. The Germ Theory of Disease requires no math

The hypothesis that germs cause disease requires no math. But using that hypothesis to make predictions, and checking those predictions against data, does. The Germ Theory of Disease is all of those things; it's not just the hypothesis by itself, any more than the General Theory of Relativity is just the hypothesis that spacetime is curved, and nothing else.

 

[BWV]

The hypothesis that germs cause disease requires no math. But using that hypothesis to make predictions, and checking those predictions against data, does. The Germ Theory of Disease is all of those things; it's not just the hypothesis by itself, any more than the General Theory of Relativity is just the hypothesis that spacetime is curved, and nothing else.

I disagree, the Germ Theory of Disease (or the Theory of Evolution, for that matter) can be fully expressed and defined without recourse to math, and can be supported by evidence with no math, except maybe some simple statistics. GR, as far as I am aware, is defined by the field equations not by a vague statement about space being curved

 

[PeterDonis]

I disagree, the Germ Theory of Disease (or the Theory of Evolution, for that matter) can be fully expressed and defined without recourse to math, and can be supported by evidence with no math, except maybe some simple statistics.

"Some simple statistics" is math. And it isn't anywhere near as simple as you seem to think it is. Nor is that the only math involved. I think you are greatly oversimplifying what these theories actually say and how they are actually used to make predictions.