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

[BillTre]

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.

 

[BWV]

"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.

but the argument you make in the OP is:

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.

and

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.

My seemingly uncontroversial point is some theories outside of physics are defined qualitatively, not by math. If you quibble with germ theory, then there is evolution, cell theory, etc. While any natural language description of GR is incomplete, but you can completely outline germ theory or evolution this way. The fact that math is somehow used somewhere with germ theory is irrelevant to the argument you make in the OP

 

[PeterDonis]

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.

 

[jackwhirl]

@PeterDonis I think that you've got the beginnings of a good article here, but that it needs some work, because it isn't clear who your target audience is, or that you've crafted a persuasive argument.

Audience:
Is your intention to refer future troublesome posters to this article to bring some fault to their attention? -- This seems most likely from the context and the general sense of agreement with @phinds that, "we'll get a lot of use out of" it. --

I ask, because the tone of the article reads like an (admittedly polite) rant about troublesome posters and most of your positive feedback is coming from people who commiserate because they also deal with troublesome posters. It seems to be written more with that audience in mind, rather than the first. So, who are you trying to convince?

Thesis/support:
And what are you trying to convince them of? 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. The article spends a lot of words talking about understanding the math, and very few about opinions and entitlement and what you mean by those words. It is also, frankly, indefensible. An opinion is "a view or judgment formed about something, not necessarily based on fact or knowledge." (Google Dictionary, 1st definition)

Most of the article is about how "natural language descriptions" don't tell the whole story. Besides the quote from Feynman -- which is underdeveloped, you could do a lot more there -- I don't see anything that supports the "opinion" half of your argument. I would expect more about what constitutes a "valid opinion" which you touch on in the comments. Maybe links to articles on logical fallacies. I'm pretty sure one of the hypothetical statements you covered would be considered an "Argument from ignorance".

Your conclusion paragraph shows the promise of a better article. It is about managing expectations, and delivering bad news. ("Sorry, but if you want this to all make sense, you'll have to put in the hard work.") This is a much better take than the path the thesis sets you on (that you promptly and properly abandon.) It is also likely to be better received and less likely to make people defensive and belligerent.

So, who is your audience, and what are you really trying to tell them?

 

[PeterDonis]

it isn't clear who your target audience is, or that you've crafted a persuasive argument.

The target audience is all PF members. The article is not trying to persuade anyone of anything.

 

[BWV]

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.

Of course you can. Math is not an important aspect of many theories, particularly in biology
So I guess Darwin is not entitled to an opinion on his own theory?

I attempted mathematics [at Cambridge University ], and even went during the summer of 1828 with a private tutor (a very dull man) to Barmouth, but I got on very slowly. The work was repugnant to me, chiefly from my not being able to see any meaning in the early steps of algebra. This impatience was foolish, and in after years I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics; for men thus endowed seem to have an extra sense. But I do not believe that I should ever have succeeded beyond a very low grade.

Charles Darwin Autobiography (p. 58 of the 1958 Norton edition)

 

[PeterDonis]

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. 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.

 

[PeterDonis]

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.

 

[BWV]

But again this contradicts the OP, which did not argue that math just has a part, you said math IS the theory:

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.

This may be true for GR but not for Evolution. The verbal description is the theory of evolution and the vast majority of the evidence is qualitative, not quantitative. Math is just not as important in that discipline, not that it does not have a role.

 

[PeterDonis]

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.

 

[BWV]

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.

That is a very different argument than saying the Theory IS the math, which again is true for GR, I think you are undercutting your own argument here by lumping quantitative physical theories with qualitative biological ones

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

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

 

[BWV]

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.

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

 

[PeterDonis]

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.

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

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.

 

[russ_watters]

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.

 

[Nik_2213]

Slightly tangential, I've run into science-deniers who spout stuff like 'you cannot trust science / math because it claims bees cannot fly'. You can tell them and tell them it only meant that era's best (static) aerodynamic model was visibly wonky. Later, IIRC, vortex shedding etc (dynamic) was recognised and factored in. But no, they can't accept the field has moved on.

Like Astronomy evolved from Earth-centered to Solar-centered. And elliptical, too. Then frame-dragging etc...
Like the devout Victorians who first claimed the existence of 'recent' sea-bed material high on a Welsh mountain was totally definitive proof of Noachian Flood. And, yes, is still claimed as such by sundry Evangelicals. And never mind that their beloved 'global flood' was blatantly plagiarised from the much older 'Gilgamesh' reportage of an uncommonly dire Mesopotamian river flood. Or that their shelly deposit turned out to have been 'bulldozed' from Irish Sea by a South-spreading glacier during ice-age low-stand...
{Facepalm...}

 

[Klystron]

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.

If memory serves, you paraphrase mathematician and popular author Lewis Carrol (Charles Dodgson) from the superb novel "Alice's Adventures in Wonderland".

Carrol taught logic and mathematics and tried to capture the slippery nature of natural language compared to mathematical beauty in his stories. I often thought of Alice's conversations with the Queen when tackling a new subject in advanced mathematics.

 

[Dale]

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.

 

[BWV]

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.

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

Many scientists believe that there is a uniform, interdisciplinary method for the practice of good science. The paradigmatic examples, however, are drawn from classical experimental science. Insofar as historical hypotheses cannot be tested in controlled laboratory settings, historical research is sometimes said to be inferior to experimental research. Using examples from diverse historical disciplines, this paper demonstrates that such claims are misguided. First, the reputed superiority of experimental research is based upon accounts of scientific methodology (Baconian inductivism or falsificationism) that are deeply flawed, both logically and as accounts of the actual practices of scientists. Second, although there are fundamental differences in methodology between experimental scientists and historical scientists, they are keyed to a pervasive feature of nature, a time asymmetry of causation. As a consequence, the claim that historical science is methodologically inferior to experimental science cannot be sustained.

http://ecee.colorado.edu/ecen5009/Resources/Cleland01.pdf

 

[Dale]

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.

 

[BWV]

[Moderator's Note: Thread spun off from previous thread in General Discussion since it is more specifically about particular scientific theories and how to test them against data.]

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

[Moderator's Note: Further discussion of the subthread related to the above link and the evolution topic has been moved to a new thread:

https://www.physicsforums.com/threa...-data-evolution-and-blood-circulation.994788/]

But this is diverging from the original argument I was objecting to, which is the physics-centered view that math is the 'language' of all scientific theories and one cannot understand fully or have an opinion on the theories without being deep in this underlying mathematical language. This is is true for GR or QM, but not evolution. Evolution does not have some underlying mathematical structure, it is more analogous to a legal argument, summing an enormous amount of observational evidence - while there is important math and experiment used, but they do not comprise the entirety of the argument and theory.

 

[russ_watters]

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.

 

[BillTre]

[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

 

[PeterDonis]

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.

 

[BillTre]

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.

 

[PeterDonis]

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.

 

[PeterDonis]

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.

 

[russ_watters]

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.

 

[Jarvis323]

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.

The target audience is all PF members. The article is not trying to persuade anyone of anything.

And you are, of course, entitled to your opinion. 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.

This is not a substantive response. I have been making specific arguments, and so have others. Vague complaints are not a valid response to specific arguments.

I'm confused. So this is a thread about a vague (not necessarily) persuasive opinion about what a valid opinion is? And now we are trying to make it clear enough to have a debate about it with valid arguments?

 

[PeterDonis]

So this is a thread about a vague non-persuasive opinion about what a valid opinion is? And now we are trying to make it clear enough to have a debate about it with valid arguments?

Please read my posts in context. The last quote you gave was in a particular subthread, in which you are not currently participating, about a particular scientific theory being discussed (evolutionary theory). The other quotes were in response to you in a different subthread which was about the topic of the article as a whole.

 

[PeterDonis]

I'm confused.

So am I. Do you have particular points to make and you're not sure whether they are appropriate to make in this thread? If that's not the case, I'm not sure what the point of your remarks is.