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 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 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 seems 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 the 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 pictures 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 a 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 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

104 replies
Newer Comments »
  1. PeterDonis says:
    He was in that small part.

    The summary I quoted, as noted, was Harvey’s own summary of the substance of his findings, and the math on the amount of blood wasn’t just a "small part". It was the essence of his argument for his conclusion, and what enabled him to refute the competing hypothesis he refuted. See below.

    The conclusions about the blood flow are the point here. Direction of flow, effects of blocking flow, etc.

    Those weren’t conclusions, those were observations. They were part of the evidence he was using, not conclusions he was drawing.

    The conclusion was that the amount of blood that was observed to be flowing in all those ways could not have been supplied continuously in such quantity from outside the body, by digestion of food–which was the hypothesis his contemporaries held, as noted in the article. He also, as the article notes near the end, used mathematical data to show that that amount of blood could not be consumed continuously by the tissues, as his contemporaries believed. Therefore, the blood must be circulating within the body. Confirming that conclusion, and falsifying the competing hypothesis, required math. And that is exactly the kind of thing I was talking about in the article.

  2. russ_watters says:
    I feel like the search for vague yet specific examples of qualitative science has at best inadvertently conceded the point. If 500 years ago we could have said "species evolve" or "apples fall" and called them scientific predictions, that doesn’t provide much value or hold much relevance to the modern scientific process. Today we model/prove such things with math/mathematical logic.

    Also, I’ll claim that the value of the "softer" sciences is still best expressed/proven with math. Behavioral sciences come to mind, there. The fact that they are often practiced without it is more a bug than a feature.

  3. PeterDonis says:
    It is about a the existence of a hypothesized ancestor.

    Ok, so how does finding a single shared trait prove that that organism is a common ancestor?

    Answer: it doesn’t. You have to look at how many shared traits there are and how that matches up with your expectations based on how long ago the supposed ancestor lived and what selective pressures each lineage was subjected to in between.

  4. BillTre says:
    A cladistical analysis is highly mathematical. You are sorting objects into heirercial categories based on closeness measured in terms of numbers of shared traits. I mean that is math on several levels. Sorry, this one may have just sunk your claim altogether.

    Today cladistic analyses are highly mathematical.
    At earlier times not so. The most primitive formal cladistic analyses were done by see where traits fell on proposed trees, and only with stable (non-homeoplastic traits-those that don’t change much and better reflec tree structure.
    Prior to the formalization of cladistics, is was similar but much less rigorous.

  5. Dale says:
    If analyzed cladistically (which is evolutionary thinking formalized) then its does not require any frequencies other than something is present (0 or 1; if you really need a number) or it isn’t. Its not about the frequencies, its about traits being found in one part of the tree but not in others.

    A cladistical analysis is highly mathematical. You are sorting objects into heirercial categories based on closeness measured in terms of numbers of shared traits. I mean that is math on several levels. Sorry, this one may have just sunk your claim altogether.

    Please explain where this comes from and what equation is supposed to mean. I don’t understand your jargon.
    It’s a general forum.

    This is derived from Bayes theorem which is a cornerstone of all inductive reasoning with uncertain information. In words it says that an observation only considered to be evidence for a hypothesis if the probability of the observation given the hypothesis is greater than the probability of the observation given not the hypothesis. That quantity is also known as the likelihood ratio or the Bayes factor.

    This is the very basis of inductive reasoning which is, in turn, the core of science.

  6. BillTre says:
    The theoretical prediction you are discussing in this particular example, as I understand it, is based on the hypothesis of common descent, which is part of the overall hypothesis of evolution by natural selection.

    This is wrong.
    It is about a the existence of a hypothesized ancestor.
    I don’t know why you have such difficulty believing this is what I am talking about.
    The existence of a hypothetical ancestor does not depend upon the mechanisms that underlies evolutionary changes.
    You keep arguing that I am talking about something else, despite my frequent protests.

    Not all evolutionary questions are answered through population dynamics.

    Looks like Harvey was making a quantitative claim based on math to me.

    He was in that small part.
    The point is that he used a lot of other pieces of evidence for this.
    Sorry to disappoint you, but I am not claiming no math is ever involved, but I am claiming that math was not required for every conclusion. I have tried to make that clear several times.
    The conclusions about the blood flow are the point here. Direction of flow, effects of blocking flow, etc. Math not required.

  7. PeterDonis says:
    You are also not explaining how math was required to determine blood circulation.

    See, for example, this article:

    https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2776239/

    Quoting from the article (emphasis mine):

    In Chapter 13, Harvey summarized the substance of his findings: "It has been shown by reason and experiment that blood by the beat of the ventricles flows through the lungs and heart and is pumped to the whole body. There it passes through pores in the flesh into the veins through which it returns from the periphery everywhere to the centre, from the smaller veins into the larger ones, finally coming to the vena cava and right atrium. This occurs in such an amount, with such an outflow through the arteries and such a reflux through the veins, that it cannot be supplied by the food consumed. It is also much more than is needed for nutrition. It must therefore be concluded that the blood in the animal body moves around in a circle continuously and that the action or function of the heart is to accomplish this by pumping. This is only reason for the motion and beat of the heart."

    Looks like Harvey was making a quantitative claim based on math to me.

  8. PeterDonis says:
    Your interpretation is wrong.

    Apparently I haven’t made it clear enough what the issue is. See below.

    I am talking about in what branch of a phylogenetic tree is a trait found in the phylogenetic record.

    What theoretical prediction is this supposed to be confirming? The theoretical prediction you are discussing in this particular example, as I understand it, is based on the hypothesis of common descent, which is part of the overall hypothesis of evolution by natural selection. That hypothesis does not predict simple yes/no answers for which traits we should find where in organisms; more precisely, just yes/no answers are not sufficient to confirm the hypothesis. The hypothesis predicts, roughly speaking, that the number of shared traits between two organisms should vary based on how long ago their last common ancestor lived and how much selection pressure each lineage was subjected to since diverging from the last common ancestors. That is the sort of comparison with data that needs to be done to confirm the hypothesis, and it cannot be done without math.

  9. BillTre says:
    I’ve already responded to this. You are greatly oversimplifying what the actual theoretical predictions are and how they have to be tested.

    Your response seems to lack any understanding of what it is to consider trait distributions on a phylogenetic tree.
    Try reading this: Phylogenetic Systematics.

    The only way you can make that argument with the cladistics example, is if you require to frequencies to replace any determination of whether something is present or not.

    Go back and read my post #59.

    which would be this I guess:

    No, the frequencies of traits are not irrelevant, because the hypothesis to be tested is that traits evolve due to natural selection. That is a claim about the frequencies of traits and their relationship to selective pressures. It is not a claim about the simple presence or absence of traits.

    Your interpretation is wrong.
    You seem to be thinking of studies on changes in populations frequencies in short term studies. Yes numbers used there.

    However, that is not my counter-example which you are claiming to have refuted (I get t choose my counter-example, not you).
    Stick to the subject your own self! Don’t try to turn my example into yours.
    I am talking about in what branch of a phylogenetic tree is a trait found in the phylogenetic record.
    It is a yes or no answer. These are completely different things.
    Talk about distributions on phylogenetic trees, not fractions of a population (which is only poorly know at best for most kinds of fossils. That is what my example is about and you seem to be ignoring it.

    You are also not explaining how math was required to determine blood circulation.

  10. PeterDonis says:
    With math not required, as I have already pointed out:

    I’ve already responded to this. I think you are greatly oversimplifying what the actual theoretical predictions are and how they have to be tested.

    You should read some of my posts on evolution.

    I’ll take a look.

    You should also read some history of science (perhaps beyond physics).

    Thanks, I’ve read plenty.

    You should also read The Origin of Species (1859).

    I have. I’ve also read Descent of Man, not to mention a lot of books on evolutionary theory published in the intervening years.

    It might give you a better idea of what evolutionary theory is about.

    I think you need to consider the possibility that your own understanding needs some work. Quite possibly mine does too, but telling me to read things I’ve already read won’t help with that.

  11. BillTre says:
    With math, yes, as @Dale has already pointed out.

    With math not required, as I have already pointed out:

    I think you miss the point.
    The phylogenetic tree is the hypothesis.
    It predicts ancestors with particular traits.
    These can be found in fossils, fulfilling a prediction.
    The frequencies at which they are found are irrelevant if one solid case is found.

    So far you have failed to give any case which can be done math free; every example you have proposed has been refuted.

    Only if you want to not read, understand, or consider what I am writing, maybe so.

    The only way you can make that argument with the cladistics example, is if you require to frequencies to replace any determination of whether something is present or not.
    On its face, an absurd interpretation. o0)

    Similarly for the example of the discovery of blood circulating.
    Blocking a blood vessel, resulting in build up of blood volume or lose of peripheral pulse.
    These are not quantitative results. They are something that happens or it doesn’t.
    Or, I guess it had to be converted to a frequency of 1 or 0 before it could be compared in a scientific manner.
    ??? o0)

    Your view of what "evolutionary theory" consists of appears to me to be much too simplistic.

    You should read some of my posts on evolution.
    You should also read some history of science (perhaps beyond physics).
    You should also read The Origin of Species (1859).
    It might give you a better idea of what evolutionary theory is about.

  12. PeterDonis says:
    If LIGO did not find gravitational waves, would that have proven they don’t exist?

    No, because we already had other indirect evidence for gravitational waves: observations of the orbital parameters of binary pulsars changing over decades, for example. Those observations were consistent with GR predictions for how orbital parameters of such systems should change over time due to gravitational wave emissions. If those observations had been different, GR’s model of such systems would have been falsified.

    And the comparison of observations with predictions of course required math.

  13. PeterDonis says:
    The frequencies at which they are found are irrelevant if one solid case is found.

    No, the frequencies of traits are not irrelevant, because the hypothesis to be tested is that traits evolve due to natural selection. That is a claim about the frequencies of traits and their relationship to selective pressures. It is not a claim about the simple presence or absence of traits.

    Your view of what "evolutionary theory" consists of appears to me to be much too simplistic.

  14. PeterDonis says:
    What I wrote about analyzing phylogenetic trees using methods like cladistics in fact can do this.

    With math, yes, as @Dale has already pointed out.

    This is not a straw man claim.

    The claim you are making now, which appears to be simply that some scientific theories can so have their predictions compared with data without using math, is not responding to a straw man claim, no.

    But you have shifted your ground from what I quoted from you before. Before, you were claiming that we are saying nobody can do any science at all without using math. "Do any science at all" is much broader than "compare the predictions of a scientific theory with data". Nobody has made any such claim, so that claim is a straw man. If you are no longer saying anyone has made such a claim, good.

    It can certainly be done math free (in simple cases).

    So far you have failed to give any case which can be done math free; every example you have proposed has been refuted.

  15. BillTre says:
    We are saying that, in order to compare the predictions of a scientific theory with data and thereby establish whether the theory is falsified or confirmed, you need to use math.

    What I wrote about analyzing phylogenetic trees using methods like cladistics in fact can do this.

    Please focus discussion on that specific claim instead of responding to straw main claims that nobody is actually making.

    This is not a straw man claim.
    It can certainly be done math free (in simple cases).
    You seem to be ignoring it, but it directly refutes this claim.

  16. PeterDonis says:
    Do you really want to restrict all science to be dependent upon using math?

    Nobody is saying that any time anyone does any kind of science, they have to use math.

    We are saying that, in order to compare the predictions of a scientific theory with data and thereby establish whether the theory is falsified or confirmed, you need to use math.

    Please focus discussion on that specific claim instead of responding to straw main claims that nobody is actually making.

  17. BillTre says:
    [Moderator’s Note: Most of the content in this post has been moved to a new thread:

    https://www.physicsforums.com/threa…ion-and-blood-circulation.994788/post-6404335

    Moved content has been deleted below.]

    Hmm, I think you are confusing feasibility with falsifiability. For example, even before the technology to detect gravitational waves was developed, the existence of gravitational waves was a falsifiable prediction.

    If LIGO did not find gravitational waves, would that have proven they don’t exist? (I am not convinced, maybe it wasn’t sensitive enough).

    You appear to not recognize that different models make different specific predictions about the energy range of a given particle. If they do not see it in a particular range then yes they will continue looking elsewhere, but that non-observation already eliminates some of the theories.

    Do you really want to restrict all science to be dependent upon using math?

    • Not all data has to be presented as math,
    • Not all advances in science have been driven strictly by hypothesis testing
    • There are inductive methods that don’t require (but of course can use) math
  18. Dale says:
    I will give counter-examples that show the these rules don’t apply.

    Sorry, but your counter examples are not very convincing.

    an important mechanism driving the changes was natural (or artificial selection (as opposed to drift or other mechanism of evolutionary change in populations)) … this line begot this line and that line by the year 1835 and in turn that line begot this other one after a new variant arose in 1845.

    I am not sure that this observation can be considered evidence for the theory. First, its non observation would not falsify the theory. Second, its observation does not distinguish between the proposed natural/artificial selection mechanism and other alternative mechanisms. So I don’t think it is actually evidence.

    Now, your characterization of these records could be incorrect, and the records could contain information about trait frequency and selection procedures. Then you could observe that the frequency of a selected trait increases while frequencies of other traits do not. That would be evidence, but it would also be quantitative. If the frequency did not change that would falsify the theory, if the frequency decreased of a selected-for trait then that would also falsify it. Etc.

    For some observation, ##O##, to be evidence for a hypothesis, ##H##, then $$\frac{P(O|H)}{P(O|\lnot H)}>1$$ and I don’t think that can be claimed with this. That calculation may not be performed explicitly, but it is the core of what constitutes scientific evidence.

    Darwin sought out animal breeders whose history and records support both evolution (change of traits in a population) occurring as well as the ability of selection to change the frequency of those traits in a population

    This seems more reasonable evidence, but this is quantitative. Frequencies can increase or decrease or vary randomly or stay constant. That is all math.

    Darwin drew the first phylogenetic tree and using his awareness of this way of thinking’s obvious predictions of common ancestors and their "intermediate" characteristics, predicted the existence of these ancestral species with "intermediate" traits.

    The phylogenetic trees are not observations. Fossil records are observations, but again, these can be quantitatively expressed in terms of frequencies of traits.

    These are predictions that although not (in the short term) of the falsifiable type

    Hmm, I think you are confusing feasibility with falsifiability. For example, even before the technology to detect gravitational waves was developed, the existence of gravitational waves was a falsifiable prediction.

    They predict some particle with properties in some range of values, build a mega-colilder, and look for positive results.
    If its not found there, look at a different energy range.
    Keep looking until you find it

    You appear to not recognize that different models make different specific predictions about the energy range of a given particle. If they do not see it in a particular range then yes they will continue looking elsewhere, but that non-observation already eliminates some of the theories.

    In any case, I am unsure why you brought that up since it really doesn’t seem to support your claim at all.

  19. BillTre says:
    Rather than argue about the poorly defined need for math to make something scientific, I will give counter-examples that show the these rules don’t apply.

    In order to have a scientific theory it must be falsifiable. That means that you must be able to make specific predictions about the outcome of experimental measurements whose result will either validate or falsify a theory. This is central to the scientific method. I see no way to do that without math.

    Evolution was not a scientific theory in that sense for quite some time after it was developed. When experimental evolution was developed it most certainly used math to make quantitative experimental predictions that were then compared to the measured outcome of actual experiments. It took time for the theory of evolution to reach that point and it is perfectly reasonable to say it wasn’t a "real" theory (meaning a scientific theory) until that point.

    I see no problem with that. Evolution is like "gravity" – it is primarily a name for an observed phenomena in nature. Darwin noticed the phenomena exists and came up with a partial explanation for how it worked. Which is great, but still limited.

    Darwin had two different major points in his book.
    For those who doubt science can be done without science, The Origin of Species is a good read, (or Darwin for wikipedians).
    One point was that change in biological forms occurs over time (Evolution occurs and is real).
    The other was that an important mechanism driving the changes was natural (or artificial selection (as opposed to drift or other mechanism of evolutionary change in populations)). Providing a mechanism made the existence of the change more plausible.

    Although experiments in long term historical sciences are not always immediately rewarded, they did exist, and observations on selection existed, even back then.

    In trait transmission to offspring and the ability to modify those traits: Darwin sought out animal breeders whose history and records support both evolution (change of traits in a population) occurring as well as the ability of selection to change the frequency of those traits in a population.
    Genetics was not at that time a mature science. Ideas of genes and how they were transmitted from parent to offspring were not clear.
    There was no biochemistry.
    Cell theory was in the process of being established.
    Evolution was based on observations of traits (many which were not well defined either) and analyzing how they changed over time.

    Darwin’s hypothesis of natural selection was a reasonable mechanism based on what was known to provide a mechanism for evolutionary change.
    The most obvious expression of this at his time was the artificial selection "experiments" of breeders.
    There were not you modern breeder’s records with detailed records of frequencies of traits of each generation so that the quantitative rate of change in a population could be determined.
    They were probably more like: this line begot this line and that line by the year 1835 and in turn that line begot this other one after a new variant arose in 1845.
    Much can be done with qualitative differences alone.

    Darwin drew the first phylogenetic tree and using his awareness of this way of thinking’s obvious predictions of common ancestors and their "intermediate" characteristics, predicted the existence of these ancestral species with "intermediate" traits.
    This is similar to Mendeleev’s predicting new elements in the holes is his table.
    A good and seemingly unlikely prediction that takes a while to be fulfilled, but provides strong support if found.
    If you are able to generate a seemingly "out of the blue" prediction, even though it may not be formally falsifiable, if found, it demonstrates a great ability for the idea to make valid predictions.

    These are predictions that although not (in the short term) of the falsifiable type, but they provide strong support for the source of the prediction.
    This is also similar to predictions of new particles by physicists.
    They predict some particle with properties in some range of values, build a mega-colilder, and look for positive results.
    If its not found there, look at a different energy range.
    Keep looking until you find it or perhaps later when you get enough negative evidence that people give up looking, or a different contrary result invalidates the search.
    It uses math, but its not invalidation activity.
    However, most people would call it science.

    An associated science from that time (which was important to Darwin’s thinking), geology did not require much math (except prehaps for mapping purposes).
    At that time, the ideas of gradualism were in conflict with the biblically inspired castrastrophism.
    Demonstrating the the significance of sedimentary layers, the great age of geological features, and finding features indicative of (as of then unknown) great prolonged forces that change the features of the land were important issues for geologists like Lyell at the time.
    The geological stratgraphic record was being compiled at this time. This would result in a series of hypotheses like, what layers should be between A and C on the other side of that hill, compared to what was found there.

  20. Dale says:
    So in your opinion, everything in the list below is quantitative, not qualitative?

    https://en.wikipedia.org/wiki/Evidence_of_common_descent

    Let’s see:

    "More closely related species have a greater fraction of identical sequence and shared substitutions compared to more distantly related species."

    "neutral human DNA sequences are approximately 1.2% divergent (based on substitutions) from those of their nearest genetic relative"

    "it can be calculated how long ago the two species diverged by …"

    I suspect that some of the observations in the list are not quantitative and cannot be made quantitative, but those observations would probably not count as scientific evidence since their non-observation would not falsify the theory. These are generally what @russ_watters was referring to as "an observed phenomena in nature".

    With evolution this issue is particularly confusing because the same name is used to talk about the theory and the observed phenomena. The theory is the part of evolution that deals with models of the mechanisms whereby the observed phenomena are produced (e.g. natural selection, sexual selection, mutation, gene transmission, etc.). To determine whether an observation is adequately explained, e.g. by mutation, it is necessary to model the mutation process and come up with a number that can be compared to the evidence. That is the theory part of evolution. Until you have that model that can be compared against the experimental evidence you have a collection of observed phenomena, not a scientific theory.

  21. russ_watters says:
    In order to have a scientific theory it must be falsifiable. That means that you must be able to make specific predictions about the outcome of experimental measurements whose result will either validate or falsify a theory. This is central to the scientific method. I see no way to do that without math.

    Evolution was not a scientific theory in that sense for quite some time after it was developed.

    I see no problem with that. Evolution is like "gravity" – it is primarily a name for an observed phenomena in nature. Darwin noticed the phenomena exists and came up with a partial explanation for how it worked. Which is great, but still limited.

    But the predictions do not have to be quantitative – they can be simple observations.

    I disagree. Even a "simple observation" can usually be made quantitative, e.g. X>0. I don’t know what sort of valid scientific evidence would not be mathematical/quantitative.

    So in your opinion, everything in the list below is quantitative, not qualitative?

    https://en.wikipedia.org/wiki/Evidence_of_common_descent

    How do you connect observations to hypothesis without at least some math? I see lots of math/implied math in that link. Let’s be specific.

    One key facet of evolution is how traits are passed along from parent to child. It’s known that the same trait can be generated independently along different evolutionary paths. So just showing that two animals have the same trait doesn’t prove they are related, much less make the relationship clear. But with genetics you can prove quantitatively how traits are passed down and figure out the actual links between species.

    The simplest (simplistic/oversimplified) common first example is eye color: One parent has blue eyes and the other brown. What are the odds of their kids having blue/brown eyes? This may be easy math, but its math nonetheless.

  22. Dale says:
    But the predictions do not have to be quantitative – they can be simple observations.

    I disagree. Even a "simple observation" can usually be made quantitative, e.g. X>0. I don’t know what sort of valid scientific evidence would not be mathematical/quantitative.

    Your physics bias is showing and you falling into the same fallacies as evolution deniers. Experimental evidence becomes very difficult with historical sciences and observation and logic become more important than numbers.

    If you do not perform an experiment then what you are doing is not the scientific method. That experiments are particularly difficult in some contexts does not mean that those contexts get a free pass and can be considered scientific in the absence of experiments.

    By the way, my background is not physics, it is biomedical engineering. I am well aware of the difficulties of making good experiments and good models and predictions in biological systems. That doesn’t mean I get a free pass either.

  23. Dale says:
    Then that leaves you in the awkward position of implying that Evolution was not a ‘real’ theory until some point years after its development and acceptance when math became a meaningful (and by what threshold?) aspect of the discipline

    I don’t find that awkward at all. I won’t merely imply it, I will assert it.

    In order to have a scientific theory it must be falsifiable. That means that you must be able to make specific predictions about the outcome of experimental measurements whose result will either validate or falsify a theory. This is central to the scientific method. I see no way to do that without math.

    Evolution was not a scientific theory in that sense for quite some time after it was developed. When experimental evolution was developed it most certainly used math to make quantitative experimental predictions that were then compared to the measured outcome of actual experiments. It took time for the theory of evolution to reach that point and it is perfectly reasonable to say it wasn’t a "real" theory (meaning a scientific theory) until that point.

    By that criterion string theory also was not a scientific theory until it could make specific experimental predictions. Nothing wrong with that, we are doing science and it takes time to properly apply the scientific method.

  24. russ_watters 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.

    I have to say this first post struck me as odd — math as a metaphor? For simplification? The way I see it, the math is used to provide a deeper and more specific understanding of the processes. I think that’s the opposite of what you expressed. However:

    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.

    If we soften that a bit, I do agree that different branches have different levels of math utilization, which means that a non-mathematical understanding may be of more value in some fields than others. But I don’t agree with your example of a disease epidemic. I think that the quantification is important enough that the superficial description holds very little value. I think our discussions of COVID show just how quickly the math becomes important. There’s not much you can discuss without it.

    Evolution is a better example, to me. The superficial description has a fair amount of power. But still, it doesn’t take long to go beyond that description. I’d say the vast majority of the threads we have on evolution require math or even exist because the OP didn’t understand or recognize the need for math.

  25. PeterDonis says:
    That is a very different argument than saying the Theory IS the math

    Which, taken in context, is not what I said. But we’re just going back and forth repeating ourselves at this point.

    You don’t consider comparing DNA sequences and evaluating the degree of similarity, for example, to be math?

    Perhaps you have a narrower view of what constitutes "math" than I do.

    that leaves you in the awkward position of implying that Evolution was not a ‘real’ theory until some point years after its development and acceptance

    No, it leaves me in the common position of using the term "theory", for the particular purpose of this article, in a narrower sense than it is often used in informal discussions.

  26. PeterDonis says:
    you said math IS the theory

    You’re quibbling. Taken in context, that statement means what I have already said multiple times in this discussion, that you need math to know what the theory’s predictions are and to compare them with data.

    The verbal description is the theory of evolution and the vast majority of the evidence is qualitative, not quantitative.

    Please give a specific example of evidence that is qualitative, not quantitative. That will be a better basis for discussion than general statements.

  27. PeterDonis says:
    I guess Darwin is not entitled to an opinion on his own theory?

    The theory of evolution as we know it today is a lot more than what Darwin presented in his books. What Darwin presented in his books left a lot of gaps, which scientists since then have spent a lot of time, and math, filling in.

  28. PeterDonis says:
    It is my opinion that, "If you don’t understand the math, you’re not entitled to an opinion about the theory" is a poor thesis statement because it is not representative.

    And you are, of course, entitled to your opinion. :wink: But note that your opinion stated here is not about any scientific theory, but about a non-scientific statement that I made–basically, my opinion, which is also not about any scientific theory. So neither of those opinions fall into the category I was talking about in the article.

  29. 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.

  30. 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.
  31. 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.

  32. 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.

  33. 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.

  34. 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.

  35. 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.

  36. 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.

  37. 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.

  38. 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.

  39. 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.

  40. 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.

  41. 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)

  42. 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.

  43. 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:

    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.

  44. 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.

  45. 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.

  46. 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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