# The problem with Mathematics in Physics

Summary:: The problem with Mathematics in Physics

The problem with Mathematics in Physics:

Consider the equation x = y x z
What does it tell us about Physics? Pretty much nothing, I am sure you would agree.

Now consider F = m x a
That's a lot better, it tells us how much force is required to accelerate a certain mass.
Great! But that's not enough, it doesn't actually tell us any Physics.
Why? Because it doesn't tell us why mass resists force. That is the Physics part.

And that unfortunately is often all mathematics can ever do.
So my parting comment to Physicists, and Cosmologists especially, is to also give us the underlying Physics and not just the math.

nuuskur, Astronuc and weirdoguy

berkeman
Mentor
Summary:: The problem with Mathematics in Physics

x = y x z
$$\frac{x}{x} = yz = 1$$

nuuskur, Mondayman, etotheipi and 6 others
fresh_42
Mentor
Summary:: The problem with Mathematics in Physics

The problem with Mathematics in Physics:

Consider the equation x = y x z
What does it tell us about Physics? Pretty much nothing, I am sure you would agree.

Now consider F = m x a
That's a lot better, it tells us how much force is required to accelerate a certain mass.
Great! But that's not enough, it doesn't actually tell us any Physics.
Why? Because it doesn't tell us why mass resists force. That is the Physics part.

And that unfortunately is often all mathematics can ever do.
So my parting comment to Physicists, and Cosmologists especially, is to also give us the underlying Physics and not just the math.
And with which letters should have Shakespeare written his sonnets in your opinion?

Shakespeare! Give us the underlying poem and not just the letters!

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russ_watters, nuuskur, Astronuc and 3 others
PeterDonis
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2020 Award
it doesn't tell us why mass resists force. That is the Physics part.

No, "why" questions are not part of physics, or indeed of any science. At least not if you expect a final answer to any of them.

For example, I could answer the question "why does mass resist force" by saying something like "because the local spacetime geometry tells the object to move along a geodesic, and the object resists any force trying to push the mass off of that geodesic trajectory". And then you would ask the obvious next question: "why does the local spacetime geometry tell the object to move along a geodesic?" (That's assuming you even accepted the answer I just gave, which is the best answer "science", in this case General Relativity, can give.) And physics has no answer to that question, other than "just because".

Even if, someday, we have a theory of quantum gravity, which can answer a question like "why does the local spacetime geometry tell the object to move along a geodesic?" with something like "because the underlying quantum gravity degrees of freedom produce effects that, in the low energy limit, look like a spacetime geometry that tells matter how to move", that still won't be a final answer, because the answer to the obvious next question, "why do the quantum gravity degrees of freedom produce those effects" will again be "just because".

And that will be true of any "why" question in any science: ultimately it will come to a point where the only answer is "just because".

Borek, Astronuc, pinball1970 and 4 others
anorlunda
Staff Emeritus
One of my favorites, Leonard Susskind, likes to say that physicists are not interested in truths, they are interested in things that are useful.

Useful to make predictions that can later be verified or refuted by experiment.

Edit: Then engineers use the useful physics to build things useful to ordinary folks.

Klystron, Vanadium 50, pinball1970 and 1 other person
Dale
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2020 Award
Summary:: The problem with Mathematics in Physics

But that's not enough, it doesn't actually tell us any Physics.
Why? Because it doesn't tell us why mass resists force.
I agree that that’s not enough, but I strongly disagree about what else is needed. The additional thing that is required is a mapping from the symbols to experimental measurements.

The end result of the scientific method isn’t a bedtime story about why mass resists acceleration, it is a theory that accurately predicts the outcome of physical experiments.

russ_watters, pinball1970, phinds and 5 others
fresh_42
Mentor
The end result ... is a theory that accurately predicts the outcome of physical experiments.
... in which an observed quantity is measured. A measurement is a comparison with some sort of scale. Hence the prediction is a machinery which maps input quantities to output quantities. In order to know the input accurately, we need a measurement of its parameters again. Thus physics is a machinery which turns one number into another number, an algorithm.

We can stack principles, put in assumptions, but even if we do not know the exact algorithm like in cosmology, we at least feed a computer with those principles and assumptions, and compare the outcome of again an algorithm with the outcome called our universe.

If we drop the requirement of a mathematical description, we inevitably drop measurement, and all algorithms, too, be it a theoretical model like string theory or a huge neural network like a simulation. If string theory will turn out to be a valid description of observables, then it doesn't matter whether it was born as mathematics. Both fields have always been and will always be cause and effect at the same time for each other. ##F=ma## connects a spring, a scale and a clock. These devices are useless without the theory ##F=ma## and the formula is useless, unless we have the devices to verify it.

Dale
No, "why" questions are not part of physics, or indeed of any science. At least not if you expect a final answer to any of them.

For example, I could answer the question "why does mass resist force" by saying something like "because the local spacetime geometry tells the object to move along a geodesic, and the object resists any force trying to push the mass off of that geodesic trajectory". And then you would ask the obvious next question: "why does the local spacetime geometry tell the object to move along a geodesic?" (That's assuming you even accepted the answer I just gave, which is the best answer "science", in this case General Relativity, can give.) And physics has no answer to that question, other than "just because".

Even if, someday, we have a theory of quantum gravity, which can answer a question like "why does the local spacetime geometry tell the object to move along a geodesic?" with something like "because the underlying quantum gravity degrees of freedom produce effects that, in the low energy limit, look like a spacetime geometry that tells matter how to move", that still won't be a final answer, because the answer to the obvious next question, "why do the quantum gravity degrees of freedom produce those effects" will again be "just because".

And that will be true of any "why" question in any science: ultimately it will come to a point where the only answer is "just because".

But, at each level in the hierarchy there is some answer using axioms from the one above it. If we never ask why and recurse up the tree, then we might as well just say chemistry, biology, psychology, sociology, etc. is just as much 'physics' as physics. Physics is supposed to be the science that does go further and further up the tree as far as possible, trying to reach the most fundamental level of explanation we can. Somewhere up there, we must have some axioms, and at that level, it is true that we can no longer scientifically offer a further explanation, but at that point, at least we should have an explanation why not. I agree in this sense with the OP, that Physics needs more than just some arbitrary math that predicts something. It needs to connect that with the larger body of work that is trying to explain things from those more fundamental axioms.

Dale
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2020 Award
but at that point, at least we should have an explanation why not
That doesn’t make sense to me.

PeterDonis
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at each level in the hierarchy there is some answer using axioms from the one above it.

Not at each level, no. That's my point. There is always some level at which there is no "level above", and no other answer to the "why" questions than "just because".

Also, the answers you refer to, from the level "above" a given level, aren't necessarily what someone who asks a "why" question is looking for. Consider, for example, the question "why does mass resist force" and the answer I offered. Is it really an answer? It explains how General Relativity accounts for the Newtonian phenomenon of "mass resisting force", but does it explain why? There are plenty of people who would say it doesn't.

That doesn’t make sense to me.
I mean, we shouldn't stop searching for deeper explanation until we know that it's fruitless, like is done in QM for example.

Dale
Not at each level, no. That's my point. There is always some level at which there is no "level above", and no other answer to the "why" questions than "just because".

Also, the answers you refer to, from the level "above" a given level, aren't necessarily what someone who asks a "why" question is looking for. Consider, for example, the question "why does mass resist force" and the answer I offered. Is it really an answer? It explains how General Relativity accounts for the Newtonian phenomenon of "mass resisting force", but does it explain why? There are plenty of people who would say it doesn't.
I don't disagree with you. And in fact, maybe each time we try to explain things from lower level principles, it is still going to be math and experiment, or axioms. But for me, leaving something at one instance of a higher level mathematical model, without connecting it to the lower levels, is not enough to satisfy my notion of what physics is about.

PeterDonis
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2020 Award
we shouldn't stop searching for deeper explanation until we know that it's fruitless

How would we know?

How would we know?
Maybe we couldn't in some cases. Then we just keep trying forever.

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So my parting comment to Physicists, and Cosmologists especially, is to also give us the underlying Physics and not just the math.

I bet you say that because you don't understand this math.

Dale
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2020 Award
maybe each time we try to explain things from lower level principles, it is still going to be math and experiment, or axioms
No maybe about it. Those are the tools of the scientific method. The ultimate level will always be explained by “because it agrees with experiment”.

I do think that the pursuit of the next level of understanding is good, but if you want to use science then you must recognize and accept what the scientific method does. The scientific method only compares theories to experiment. It cannot do anything else.

I do think that it is a bit odd that @Tanelorn was complaining about the math when the math is exactly the thing that connects different theories and allows one theory to be explained in terms of a more fundamental theory. It is kind of “biting the hand that feeds” to complain about math and demand “why” when the only sort of “why” science can answer is through the mathematical connection between theories.

etotheipi
Homework Helper
Gold Member
2020 Award
There's a funny (although maybe by now a bit clichéd) video on youtube with Prof Feynman talking about Aunt Minnie slipping on the ice and breaking her hip, and then being taken to the hospital by her husband because he's interested in her welfare.

That was in response to being asked something about magnets? Worth a watch, haha.

Astronuc and pinball1970
Well one thing F=ma tells us is that 2000 years of Aristotelian thought was incorrect when it posited that motion requires a sustaining force. How valuable is that insight? Priceless, IMO.

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Mark44
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Astronuc and etotheipi
Mark44
Mentor
Consider the equation x = y x z
Presumably you mean ##x = y \times z##. If so, it's very bad form to use 'x' for a variable as well as to indicate multiplication.

What does it tell us about Physics? Pretty much nothing, I am sure you would agree.
Yes, I agree, but as others have already stated, equations such as ##F = ma## or ##\tau = I\alpha## tell us a lot about the force that results from accelerating a mass m by an acceleration a, or the torque that results from a rotational acceleration ##\alpha## of an object with a moment of inertia I.
Great! But that's not enough, it doesn't actually tell us any Physics.
Why? Because it doesn't tell us why mass resists force.
No, as has been amply explained in this thread, and explained by Richard Feynman in the interview I linked to in my previous post.

Astronuc
PeterDonis
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2020 Award
Not the whole story, but not entirely incorrect!
https://arxiv.org/pdf/1312.4057.pdf

While this paper does give plenty of good analysis, I can't help a brief off topic rant about a couple of items. First, this, from p. 6, just after equation (19):

Heavier objects fall faster even in the approximation when we disregard friction with the air!

But not if we also disregard buoyancy, which Rovelli just said on the previous page was negligible in comparison with the weight. Which is exactly the approximation that all those high-school books that Rovelli dismisses are using. (My high school physics teacher would have raked Rovelli over the coals for a statement like this.)

A more important issue, though, arises from this statement on p. 5:

If the reader thinks all this is “intuitive” and “self-evident”, he should ask himself if he would have been able today to come up with such an accurate and detailed account of the true motion of falling object.

My question is different: if Aristotle was so smart and such a good observer, how come he didn't come up with the idea that Galileo did? Rovelli tosses off this remark a little earlier on p. 5:

...unless one is as smart as Galileo to guess, correctly, that an incline would slow the fall without affecting its qualitative features...

So Galileo was really that smart? Smarter than Aristotle? And not just Aristotle, but everyone else who ever considered these questions for the next 20 centuries or so after Aristotle?

I simply don't buy such a claim. I think the advantage Galileo had was not that he was so much smarter than everyone for the previous 20 centuries, but that he was fortunate enough to live at just the right time to latch onto a better conceptual scheme.

Imagine what would have happened if the idea of testing a simple hypothesis by experiment had occurred to someone in Aristotle's time. (Actually, Archimedes might well have gotten there had he not been killed by stupid Roman soldiers.) We might have had modern physics and modern technology two thousand years ago. For all his skill at observation, Aristotle was still drastically limited by having to work only with the observations that were given to him, instead of having the key insight of figuring out how to create your own observations using artifically constructed experiments. And, for better or worse, that limited viewpoint was Aristotle's intellectual legacy.

etotheipi and russ_watters
fresh_42
Mentor
There's a funny (although maybe by now a bit clichéd) video on youtube with Prof Feynman talking about Aunt Minnie slipping on the ice and breaking her hip, and then being taken to the hospital by her husband because he's interested in her welfare.

That was in response to being asked something about magnets? Worth a watch, haha.

(bookmarked!)

Astronuc
PeterDonis
Mentor
2020 Award
not if we also disregard buoyancy, which Rovelli just said on the previous page was negligible in comparison with the weight.

Just to put some typical numbers to this, consider a sphere made of something like rock, falling vertically in air. The sphere has cross sectional area ##A = 0.1 \text{m}^2##; this gives a volume for the sphere of ##V = 0.0238 \text{m}^3##. Rovelli's friction coefficient ##C## is what aerodynamics engineers typically call ##C_d A##, the drag coefficient times the cross sectional area; for a typical drag coefficient of a sphere in air, something like ##C_d = 0.5##, this gives ##C = 0.05##. Then the buoyancy force is ##V \rho## and the friction force is ##C \rho v^2##, where ##v## is the speed of the sphere. The weight of the sphere, if we assume a typical rock-like density of ##2000 \text{kg / m}^3##, is ##m g = 466 \text{N}##. A typical density for air is ##1 \text{kg / m}^3##, so we have a buoyancy force of ##0.0238 \text{N}##, which is indeed negligible compared to the weight, but also will be smaller than the friction force for any ##v > \sqrt{V / C} = 0.69 \text{m / s}##. The body will reach this velocity in about ##0.07 \text{s}##.

russ_watters and etotheipi