## Physics is more than applied mathematics

I thought rather hard before reblogging this, as I do not wish to cause any conflict between the different parts of my School – the Department of Mathematics and the Department of Physics and Astronomy!

I don’t think I really agree that Physics is “more” than Applied Mathematics, or at least I would put it rather differently. Physics and Mathematics intersect, but there are parts of mathematics that are not physical and parts of physics that are not mathematical.

Discuss.

A problem set for potential applicants in the foyer of the Physics department of a premier UK university. It looks like physics, but it is in fact maths. The reason is that in the context of this problem, the string cannot pull a particle along at all unless it stretches slightly. Click the image for a larger diagram.

**While accompanying my son** on an Open Day in the Physics Department of a premier UK university, I was surprised and appalled to be told that Physics ‘*was applied mathematics*‘.

**I would just like to state here** for the record that Physics is * not*applied mathematics.

**So what’s the difference exactly?**

**I think there are two** linked, but subtly distinct, differences.

**1. Physics is a science** and mathematics is not.

**This means that physics has an experimental aspect**. In physics, it is possible to disprove a hypothesis by experiment: this cannot be done in maths.

**2. Physics is about…**

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July 15, 2015 at 2:29 pm

” In physics, it is possible to disprove a hypothesis by experiment: this cannot be done in maths.”Sure it can, and has been. People have had hypotheses which were not proved wrong in a mathematical sense, but by finding a counterexample. That certainly qualifies as “experiment”.

I could come up with some examples if necessary for ones which have already been disproved.

Take the hypothesis known as Goldbach’s conjecture: every even number is the sum of two prime numbers. It is certainly conceivable to disprove this by finding a counterexample.

I would argue almost the opposite: in maths it is possible to

provea statement to be true in an almost absolute sense, whereas in physics it might turn out to be the limiting case of a better theory.July 15, 2015 at 3:28 pm

I don’t think protonsforbreakfast’s argument holds together:

* The mathematical technique of “proof by exhaustion” is pretty obviously empirical — but of course any conjecture with an enumerable number of cases can be attacked with proof by exhaustion. I guess you can then go on to talk about Goedel incompleteness and falsifiability but we have to be very careful if we make the argument that the existence of things like the halting problem means mathematics as a whole is not scientific; it feels like doubletalk as regards to the idea of something being “inherently falsifiable” to me.

* As far as physics being “about this world” goes, we’ve all seen how many papers on ArXiv gr-qc are about something like “stability of a black ring in a 4+3 spacetime” or whatever. Plenty of what is produced by theoretical physicists is effectively math papers. Meanwhile on the other side of it — well, since mathematics is carried out with symbols that exist in the physical world, at that fundamental level all mathematics is physical (a non-Marxist might well reject this argument out of hand though).

* And finally as regards how Faraday never wrote an equation — well, to go back to the idea about symbols, writing equations is not necessary for something to be math. Euclidean proofs are often not equations in the sense that they’re written “something plus something over something equals something…”, they’re circles and straight lines. Are we really going to say that just because Faraday didn’t write out “del cross E equals negative dB/dt” that he wasn’t doing math, if the statement he made — a time-dependent magnetic field induces a non-conserved electrical field — is equivalent? In science we take individual observations and build laws as abstractions; those abstractions live in our heads in the cognitive world of symbols, the same place mathematical objects live. To say that those two sets of symbols aren’t comparable strikes me as being dependent on an overly concretized picture of what mathematics is.

July 16, 2015 at 9:20 pm

As far as the physics/maths debate goes, I’ll just say “The map is not the land” 🙂

But I think protonsforbreakfast has missed the point in calling the problem “stupid”. It’s using a classic Socratic pedagogical device: make a question subtly ill-posed, set it to the students and let them get contradictory answers, and realise they need to think deeper. Very good intro mechanics problem, IMO.

July 17, 2015 at 4:31 pm

As a mathematician I can agree that “there are parts of mathematics that are not physical” but I cannot really imagine how there could be “parts of physics that are not mathematical”. As long as they concern laws that are suppose to hold under given circumstances, I would imagine that even though these laws are not explicitly formulated mathematically they should be amenable to a mathematical formulation. Otherwise said there should be a mathematical model within which the physics becomes mathematics. Alternatively could it be possible to have a physical phenomenon that somehow contradicts a mathematical law? Apriori this should be possible, I guess… (and I am not talking here about having a bad mathematical model, that does not accurately reflect the physical reality, that’s something that often happens:).

July 17, 2015 at 4:55 pm

I am taking physics to mean “what physicists do” rather than “the physical world”. I’d say that there are two main aspects of Physics which are not mathematics. One is the creation of theories about the natural world, by which I mean the postulation of principles or hypotheses such as relativity. Although relativity became a (highly) mathematical theory it us founded on postulates which could not be derived mathematically.

The other thing about physics that sets it aside from mathematics is the role experiment and measurement play in testing theories.

Mathematics in itself can only test theories for inner consistency. Creation of theories and testing them using observed phenomena both lie outside the realm of mathematics.

July 17, 2015 at 11:29 pm

Peter, I am not convinced; your ‘two main aspects of Physics which are not mathematics’ are present too high level in what is called ‘mathematical modelling’. Let’s consider, for example, a maths model of the internal combustion engine. Any good such

mathematical model will need to strike a balance between

(1) the desire for physical realism,

(2) the extent of experimental information on the physical processes occurring in the engine, and

(3) the capabilities of today’s computers.

If we want to derive a ‘system of differential equations’ with proper parameters that correspond to processes in our engine, certainly we will confront our equations we ‘derive’ with what we can measure in experiments there.

From measurements we can even formulate hypotheses (e.g., that during combustion the pressures of burned and unburned gases are equal and their heat transfer areas are proportional to their respective mass fractions), and use this hypotheses in our derivation of the model.

So, is this ‘Physics’ for you?

July 18, 2015 at 4:16 am

I’d say it was engineering, or in other words applied physics! 😉

July 18, 2015 at 8:23 am

There are no strict/sharp boundaries between these disciplines, and some aspects

may fit better into mathematics, some into physics. For me, an engineer is

someone who can suggest/design/construct this engine, but the one who is ableto describe

physical proceses in it by mathematical equations would be mathematician/physicist.

Of course, ideally, close cooperation of all three types of expertises could bring

best results.

July 18, 2015 at 8:26 am

Exactly so. Some applied mathematics could well be called physics, but some could is biology or engineering etc. My point in the post was that there is more to mathematics than applied mathematics and there is also more to Physics than applied mathematics.

July 18, 2015 at 9:47 am

So there is physics and Physics;)

July 18, 2015 at 12:31 pm

As the case may be…

July 21, 2015 at 7:08 pm

I’m appalled that Protons for Breakfast would describe this problem as “stupid” or unphysical, because it’s one of the most celebrated problems in first-year mechanics: the one where two billiard balls collide and stick together. Sure, the sunglasses and fake moustache make it look a bit silly, but who can blame it? It’s been at the front of every mechanics textbook for hundreds of years; it probably gets mobbed every time it leaves its house.

I don’t even think the disguise is as silly as it looks. In the traditional formulation of the sticky collision problem, we imagine the billiard balls interacting through a repulsive force. In this version, the force is attractive, but the problem is exactly the same. This foreshadows the deeper result that attractive and repulsive Coulomb scattering are also essentially the same.

PfB complains that this formulation of the problem provides “no mechanism that could possibly dissipate energy,” but of course the traditional formulation doesn’t provide one either. An inelastic collision between perfectly rigid spheres is just as ridiculous as an inelastic tightening of a perfectly inextensible cord.

PfB points out that it’s unclear whether to ditch the conservation of energy or the conservation of momentum in this collision, since obviously one has to go. Again, the attractive and repulsive formulations of the problem both have the same ambiguity. I seem to recall that my high school physics teachers resolved the ambiguity essentially by reference to experiment. They noted that practical sticking interactions always seem to absorb some energy, so it makes sense to model an ideal sticking interaction as one where the momentum does what it wants, remaining constant, and the energy does what it must. A more belligerent physicist, I imagine, would just say that the choice is obvious if you think about it, and anyone who says it’s not is either an idiot or a mathematician.

At this point, I should probably turn my coat and admit that I’m a mathematician, and I agree with PfB that the attractive formulation of the sticky billiards problem is stupid, and I think the more traditional repulsive formulation is stupid too! In a cynical moment, I’d say that ambiguous models with unstated assumptions that are supposed to be obvious if you think about them are precisely what separate mathematics, where they’re frowned upon, from physics, where they’re par for the course. The divergence starts early: as an undergrad, tutoring younger students in intro mechanics, I remember being amazed at how ambiguously many of the homework problems were stated, especially problems involving constrained motion.

So, here we have a problem that looks cute and well-posed, but in fact can’t be solved without a crucial extra assumption. It looks like math, but it is in fact physics. 😉