Learn the Math

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

I’m sure anyone who has hung out long enough here on Physics Forums has encountered threads which 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 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.

[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 in 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 out. I am simply trying to distill a common argument down to what seem to me to be its bare essentials.)

What I’m about to say is going to sound harsh, and in a way it is harsh, which is why I’m saying it here instead of in any of the numerous forum threads in which I’ve been tempted to. Here it’s not directed at anybody; I’m just stating something that I think is true. Here it is:

If you don’t understand the math, you’re not entitled to an opinion about the theory.

Richard Feynman once said, “If you want to understand Nature, you must learn the language She speaks in.” It’s all very well to try to get a start by reading descriptions in English, or whatever your language of choice is, of what a scientific theory says. But those descriptions are not the theory. You can’t form an opinion about the theory from them. You have to understand the actual theory, i.e., the math.

“But scientific theories aren’t just math.”

Yes, I know that. Obviously the math is no good unless you have some way of linking the math up with experience. That’s also part of the theory, yes. But that doesn’t mean you can get away with not understanding the math, because (1) the math is what makes the predictions, and the predictions are numbers anyway, and (2) the data you’re going to compare the predictions with are numbers too, often numbers which require sophisticated interpretation before you can compare them with the predictions. And how do you do such sophisticated interpretation? With math.

I want to make it clear what I am not saying. I am not saying that scientists, and people like me who are not practicing scientists but who are knowledgeable about at least some areas of science, shouldn’t try to give clear descriptions in plain natural language of what a theory says. They should. I try to do that here on PF. But these are descriptions of what the theory says, as best it can be translated from math into natural language. The OP in my example above, and many others like him, want to demand proofs in natural language that the theory is correct, and that is just not going to happen.

“No, you’re wrong. I don’t insist on a proof. But I do insist on some explanation of what’s going on that makes sense.”

Same answer: there may not be any such explanation in a natural language that “makes sense” according to your criterion. (Very often the person making this demand doesn’t realize that what “makes sense” to them already implicitly makes a lot of assumptions that are simply not true in general, however good they may be as approximations in everyday life.) Or there may be such an explanation, but nobody has thought of it yet. It can take decades even for experts in the field to understand some aspect of a theory, and they know the math. (The history of theorists’ understanding of Schwarzschild spacetime is a good example of this, one which has given rise to a fair proportion of the threads that first gave me the idea of writing this post.)

“Yes, I know your math says X. But this other math says Y, which is inconsistent with X. And Y seems much more intuitively sound to me. So I believe Y.”

(For example, X = the proper time to the horizon for an infalling observer is finite; Y = the coordinate time is infinite.) No, Y is not inconsistent with X. To someone who understands the math, this is obvious; but if you don’t understand the math and are relying on natural language descriptions, yes, they certainly can sound inconsistent, particularly since many authors are sloppy in their terminology because they are more concerned with getting across some picture of what they’re talking about than with strict accuracy and consistency. They don’t expect what they write to be taken as a proof of the theory, or a completely consistent explanation of it, just as an attempt to describe some aspect of it in a way that is not going to be judged based on apparent consistency with other aspects.

“So you’re saying I can’t trust these authors to tell me what’s really true?”

If by that you mean “tell a completely consistent story that encompasses all aspects of the theory”, then no, you can’t. There is just no way to tell that story without the math. Here’s why: a theory is not just a description of what happens. It’s a way of generating predictions about what will happen in scenarios you haven’t looked at yet. For practicing scientists, of course, “scenarios you haven’t looked at yet” means “scenarios that haven’t yet been tested in experiments by anyone”, so a practicing physicist in General Relativity doesn’t have to spend a lot of time verifying that GR gives the correct prediction for, say, the precession of Mercury’s perihelion; he’s already been there and done that. But if you’re posting here on PF asking questions about GR, you probably haven’t been there and done that; so for you, it’s perfectly legitimate to ask how GR comes up with the correct prediction for the precession of Mercury’s perihelion (or anything else it predicts). But you can’t do that just from natural language descriptions of GR, because GR doesn’t use natural language descriptions to make its predictions; it uses math. So relying on natural language descriptions of GR to generate your predictions won’t work; you’ll be working with the wrong set of concepts.

Here’s a simple example (using SR rather than GR, but the point is the same): we get fairly frequent threads here on PF where someone is trying to figure out how “time dilation” works and getting obviously nonsensical answers. The thread will go on for many, many posts, with people trying to explain why the OP is getting obviously nonsensical answers and how they need to change how they are looking at the problem, but sometimes it just doesn’t get through. The reason is simple: the OP simply doesn’t get that “time dilation” in SR is not a fundamental concept; it’s not what the theory uses to actually generate predictions. All it is is language that some physicists use to describe what happens, *after* they’ve already made a prediction using the actual theory (the math) and verified that it’s correct.

“But then why do all those pop science books and TV shows give all those colorful natural language descriptions? Aren’t they trying to explain the theory to us?”

Yes and no. They’re trying to “explain” the theory, for some value of “explain”, yes; but if you’re asking questions here on PF, you’re probably not their intended audience. If you’re asking questions about a scientific theory here on PF, you’re already different from most people who read popular books or articles about science. Most people who read natural language descriptions of a theory don’t want a completely consistent all-encompassing story that they can use to generate predictions; they just want a quick “sound bite” that gives some flavor of what the theory says. Those are the people most of these popular authors are writing for. (There are exceptions, and I and others here on PF try to point to them where we can. Kip Thorne’s Black Holes and Time Warps is one example, a popular book that, while it can’t tell the complete story since it doesn’t include the math, manages to tell quite a lot of it without too much distortion in translation. But even that isn’t enough to “prove” that GR is “correct” or to use it to make correct predictions, if that’s what you’re looking for.)

If you’re posting here on PF, you are hopefully looking for more than just a quick sound bite or a nice-sounding description that may or may not match the actual theory. That’s great! Please post and ask questions. But don’t be fooled into thinking that we can paint a comprehensive, self-consistent picture of any scientific theory, that generates correct predictions and shows you how it’s done, without using the math. It can’t be done. So if that’s what you’re looking for, you’ll have no alternative but to learn the math. If you aren’t willing to do that, then you aren’t entitled to an opinion about the theory. You may not like it, but that’s the way it is.

Comment Thread

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  1. PeterDonis says:
    some theories outside of physics are defined qualitatively, not by math. If you quibble with germ theory, then there is evolution, cell theory, etc.

    None of these are "defined qualitatively, not by math". All of them use math to make predictions and compare them with data, and you need to make predictions and compare them with the data if you want to determine whether a theory is correct. And the discussion in the article is about what it takes to have an opinion about whether the theory is correct.

    any natural language description of GR is incomplete, but you can completely outline germ theory or evolution this way

    No, you can’t. You might think you can, but if you actually try it, you will find that you can’t. Again, you are greatly oversimplifying what it actually takes to "completely" describe the theory and its predictions and how they compare with data.

    The fact that math is somehow used somewhere with germ theory is irrelevant to the argument you make in the OP

    "Math is somehow used somewhere" is a gross misrepresentation. The fact that math is used to make predictions and compare them with data is absolutely not irrelevant to the point. See above.

  2. BillTre says:
    Concepts of geometric growth and statistics of spread and infection involve an understanding of a basic set mathematical ideas.
    These underlie the real world importance and application of Germ Theory and why it would be of interest to a more general public.
  3. PeterDonis says:
    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.

  4. PeterDonis says:
    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.

  5. PeterDonis says:
    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.

  6. PeterDonis says:
    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.

  7. russ_watters says:
    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.

  8. PeterDonis says:
    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.

  9. russ_watters says:

    Fred Wright said

    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.

  10. russ_watters says:
    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.

  11. PeterDonis says:
    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.

  12. PeterDonis says:
    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.

  13. phinds says:
    … 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)

  14. PeterDonis says:
    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.

  15. PeterDonis says:
    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:


    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.

  16. phinds says:
    — 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.

  17. PeterDonis says:
    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.

  18. phinds says:
    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".

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