Over at Crooked Timber, Daniel picks up the Harry Collins thing I talked about last week, and asks an interesting question about the role of math:

We don't want to make "understanding the subject" mean "being able to do calculations about the subject", unless we have some reason to believe that this is a necessary condition rather than a sufficient one (and to be frank, I don't believe it's a sufficient condition; I've spent enough time with economists to know that ability to do the maths does not mean that someone understands the economics). Is there anything? Or is Collins' concept of "interactive expertise" really all there is, in terms of understanding?

The question isn't unique to Daniel, of course-- physicists have come up with this one all on their own, and debated it in bars and at conferences for years. I'm not aware of any conclusive answer, but it's occasionally fun to think about. Most of the usual arguments are reproduced in one form or another in the comments to that post.

Personally, I tend to like the possibly apocryphal comment attributed to Feynman, to the effect of "if you really understand a subject, you ought to be able to explain it in terms that a college freshman can understand." Mere facility with numbers doesn't equal real understanding of a subject, and if the only way you can explain some result is by invoking higher math, there's either a problem with your model, or your understanding of it.

At the same time, though, I'm not entirely comfortable with saying that somebody who can't interpret the equations of physics really understands the subject. It's a tough problem.

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I would argue that it is not necessary to work out the mathematics to

understandphysics, but it is necessary in order todophysics.I'll just add here what I've been saying in the previous thread:

If one doesn't understand the theory quantitatively, then one doesn't understand it well enough to know

themost basic thing about the theory--does it describe reality, or not?if you really understand a subject, you ought to be able to explain it in terms that a college freshman can understand.Us software folks extend that to 'if you really understand a subject, you can program a computer to solve its problems'

So I'm a big believer in the idea that if you can't explain your research over a pint of beer, then you're a charlatan. But I also believe that if you can't quantitatively determine whether or not an unexpected new observation is in agreement with or in conflict with the theories in your field, you are also a charlatan. Both aspects of understanding are necessary.

Do you really understand Pushkin's poetry if you can't read Russian? I take it you've read the criticism and some translation and can therefore talk as though you understand it.

I believe the usual argument is that 'mathematics is the language of physics', so while obviously understanding the math does not automatically mean that someone understands the concepts, it would be difficult to understand many concepts in physics without having some mathematical 'literacy' as many of these concepts are expressed in mathematical language, for example as equations.

As a total math and physics layperson (didn't pursue past 2nd-year university) could I expand on Frumious B's point?

I think this is something like the difference between /appreciating/ music or poetry, and /creating/ music or poetry.

I enjoy math and physics on about the same level and in a similar way to how I enjoy music and poetry.

At my level of expertise, I can no more compose a symphony than I can prove Fermat's last theorem or work with string theory. But I can still think it's all really neat and enjoy listening to it/discussing it/reading about it/etc.

My limited knowledge, of both math or art, constrains my ability to understand and appreciate them at the level an expert might, but it's not a binary thing of "totally groks it" or "has no clue".

It seems to me that a so-called expert in any field who can't explain their work to a layperson so they can at least grasp the big picture - whether it's the physicist who resorts to exotic math, or the arts type who buries everything in post-modern terminology to the point that "Towards a Transgressive Hermeneutics of Quantum Gravity" sounds reasonable - isn't much of an expert.

It depends on what you mean by understanding. I would argue that there are some concepts that elude understanding on an intuitive level, a way that makes sense by describing them in words. There are obviously some concepts that are quite well understood in a nonmathematical way. I can describe various atmospheric phenomena without math, but this is mainly (or at least usually mainly) because they can be related to other concepts; in other words, understanding by analogy. But there are some concepts that really have no analogy in the day-to-day world. We resort to strained analogies or stories that seem to capture some of the flavor, but leave the listener confused. There simply are no phenomena that are even rough analogies, and, since we have no first-hand experience, we have no intuition. Some of these concepts can be understood only by the math.

I once read part of an interview with a well-known physicist (I wish I could remember more, like his name) who said that even he could not say that he could grasp some of the concepts on an intuitive level. I don't think that's a failure in his powers of understanding; for example, I am pretty sure I don't believe that a person can actually visualize more than three spatial dimensions in any true sense. If someone can do that, please show it to me without the math.

I think it's more than a bit pat to presuppose that 'easy to understand' means 'explained in English' and 'difficult' means 'uses math'. Now granted, mere symbol manipulation doesn't equate to understanding, but any

mathematicianwould agree with that truism.Specifically when it comes to english explanations at say the Brief history of time level, the universe is no more a novel than War and Peace is a topology textbook. To understand anything you must possess the necessary tools, and you don't do calculus in iambic pentameter.

What I meant to say was that it's an error to simply assume that the more "fundamental" understanding is the one that uses as little math as possible. To insist too strenuously otherwise seems to me to reduce science to science fiction.

I think being able to explain everything using just the terms of a freshman is a huge oversimplification. The key here is not necessarily calculations, but formalism.

A precise understanding of a subject requires precise definitions. As you dive deep into a subject, you accumulate more and more terminology. At some point, there are no deductions that you can make without invoking graduate level terminology (or reinventing them).

The gotcha here is that knowing mathematics doesn't automatically make you a physicist; you only know how to use the formulas but you don't know whether the model is correct. The real physicist is the one who understands nature well enough to be able to express it using formalisms.

I think being able to explain everything using just the terms of a freshman is a huge oversimplification. The key here is not necessarily calculations, but formalism.

A precise understanding of a subject requires precise definitions. As you dive deep into a subject, you accumulate more and more terminology. At some point, there are no deductions that you can make without invoking graduate level terminology (or reinventing them).

The gotcha here is that knowing mathematics doesn't automatically make you a physicist; you only know how to use the formulas but you don't know whether the model is correct. The real physicist is the one who understands nature well enough to be able to express it using formalisms.

"If one cannot state a matter clearly enough so that even an intelligent twelve-year-old can understand it, one should remain within the cloistered walls of the university and laboratory until one gets a better grasp of one's subject matter."a quote i've seen attributed to margaret mead

How much math did Margaret Mead use?

The gotcha here is that knowing mathematics doesn't automatically make you a physicist; you only know how to use the formulas but you don't know whether the model is correct. The real physicist is the one who understands nature well enough to be able to express it using formalisms.The question isn't "Is Collins a physicist?" the question is "Does Collins understand physics?" I think that one can understand physics without being able to

dophysics. And it's most definitely possible to solve problems in physics without reallyunderstandingphysics-- look at any intro class, and you'll find a few people who do a good job of manipulating formulae, but get simple conceptual questions wrong.Chad,

A clarification. The question is not "Does Collins understand physics?" Not at all. The question, as Davies has clarified in the comments, is whether Collins understands it at the same level as the people actually doing gravity wave physics, the people he's been rubbing elbows with for 30 years at conferences and visiting elsewhere and studying as material for sociological research.

Clearly Davies wants to prevent physicists from claiming that he does not -- I have thrown out a number of examples of how the mathematical tools and the formalisms are inseperable from physics, both as a tool for communicating with others and as a way of seeing further than we would without it. In fact, I cited Terrell rotation as an example of how the formalism isn't sufficient but it is necessary, and he completely missed the point. Another example I provided was the ultraviolet catastrophe and Planck reconciling Wien's law and the Rayleigh-Jeans law for blackbodies. Another was an example from my own work about coding up propagation of extraordinary mode waves (no response on that one yet). The man simply refuses to believe that physics has more to it than a sociological endeavor.

Hmmmm....

Experimental science, including "applied" physics is defined by a process that relies on logical induction, whereby even given the truth of a conclusions supporting premises, the conclusion itself cannot be said to be unconditionally true. Logical induction compares degrees of 'logical strength" or "weakness" based on the rigor of it's support and etc.

Mathematics is an example of "logical deduction". Given the truth of the supporting premises, the conclusion MUST be true as well.

It should be obvious that when a method of logical deduction is used to DESCRIBE a process that is defined as logically inductive, there's going to be a lot of confusion. Mainly because the method of description cannot claim a one to one commutability with the process it describes. "Inductive strength" is NOT the same as "Deductive truth", which is often lost between the theorists, who use mathematics to idealize physical processes, and computer programmers and engineers who have to try to puzzle out first how to accurately translate, and then subject themselves to weird hypothesis that they're expected to take remotely seriously.

Mathcad, Maple, and Mathematica... you know, that software used to knock out the undergrad physics exams, does not run on pure mathematics... perhaps before teaching students the mathematics of physics as if they were learning about physics itself, it would be wiser to teach the students enough programming to understand the algorithms and logic ACTUALLY used? Crazy thought, I know, but still... Physics is only the Queen of the Sciences. So what is the King of the Sciences?

Ahhhhh! Who cares. You'll always have engineers.

to be a practicing physicist, a solid understanding of mathematics is neccessary, but not sufficient! If nothing else you need to deal with boundary conditions, which come from physics, not math.

BUT it is neccessary, even for them pesky experimentalists.....

I've taught Conceptual Physics, as well as math-based physics, at the high school level for 20+ years now. During that time, I've hopped from one side of the fence to another several times. Here's my take on the situation now. My opinions may change by tomorrow...

It is possible for a layperson with little or no mathematical skill/talent/awareness to understand physics -- to an extent. This belief is the basis of Paul Hewitt's Conceptual Physics course and text, and it does work. Approaching Physics as an intellectual exercise, rather than a series of math problems, appeals to many humanities-oriented students. And a few actually take the next step and enroll in a math-based course later on. One of my Conceptual grads just earned a physics degree from Wheaton College (no relation, BTW), in fact.

The non-math approach can only take a student so far, though. It's little like learning to play an instrument without learning music theory. You can play guitar like Clapton, but to create music, it's really a big help to know at least some music theory. (Granted, there are composers who can write music intuitively. They are as rare as physicists who can work by the seat of their pants, I'd say.) Applying numbers to the concepts helps solidify the concepts and stresses that physics is not a set of ideas that can be endlessly debated, like (dare I say) philosophy or literary theory, but a really powerful predictive science with little room for interpretation. Math makes that point pretty concrete.

So to be an expert in physics, you need math. To appreciate it, I don't think math is necessary.

Exposure to maths at an early age provides one with a glimpse into a magic wardrobe. Open and try on whatever you like, then model it for the world.

There is a wonderful beauty and elegance to be found within the discipline, and a desperate need to encourage numeracy. A huge challenge for the scientific community is to make all topics, not just our favorites, accessible so that we expand the awareness of, and need for, a solid grounding in scientific thought. The same holds for literature and other areas.

What do you recommend to reach out to students of all ages?

IMO "Knowing Physics" isn't a yes/no thing. It is a continuous thing in many dimensions. To do experiments you must know the math because you need it to a) make quantitative predictions, and b) to analyse the results to see how they matched the predictions. To understand relationships between things you don't need the math - you can describe complicated differential equations as changes in one thing compared to changes in another. Both of these levels have the same "truth" to me, so you have to get down to what the person is attempting to do, to see if they are trying to use a qualitative relationship to make quantitative predictions or misinterpreting quantitative results in a qualitatively wrong way.