Why are Physicists so informal with mathematics?

[PeroK]

@AndreasC I beleave you use the word "rigorous" differently than the way mathematicians use it.

When I was a graduate student I worked my way through A Hilbert Space Problem Book, by Paul Halmos.

https://link.springer.com/book/10.1007/978-1-4684-9330-6

It included a series of open questions, where you had either to proof something or find a counterexample - without being given the usual undergraduate hints that it was true or not. As I recall, almost every statement that looked intuitively attractive had a counterexample; and, statements that looked like they couldn't be correct could be proved. Even the statements of the problem required a grounding in rigorous mathematics

Halmos was trying to achieve what Tao describes above: retraining the intuition of a student to be based on sound, rigorous mathematics, not on woolly thinking.

 

[martinbn]

You might be right but I think maybe the issue is that it's not really used consistently in general...

I think any two mathematicians will agree every time if something is rigorous or not.

 

[AndreasC]

I think any two mathematicians will agree every time if something is rigorous or not.

Different fields have different standards of rigor so I doubt it... After all I think sometimes there is a bit of confusion between "formal" and "rigorous"...

 

[martinbn]

Different fields have different standards of rigor so I doubt it...

You can try. Ask some and see.

After all I think sometimes there is a bit of confusion between "formal" and "rigorous"...

I was going to say that, it seems that you are thinking about "formal" not "rigorous", because Bourbaki and Poincare are rigorous, but the first is more formal in style.

 

[AndreasC]

I was going to say that, it seems that you are thinking about "formal" not "rigorous", because Bourbaki and Poincare are rigorous, but the first is more formal in style.

I see your point, however Poincare is not fully rigorous even in that sense. There are examples of theorems and proofs he published that turned out to be kind of wrong due to relying on more or less heuristic arguments, and which were later refined either by others or himself. If you read Analysis Situs for example, there are lots of things he doesn't justify. For instance he initially stated the Poincare duality in 1893, without proof. Then in Analysis Situs he "proves" it, but Heegard showed the proof was more or less wrong. If I recall correctly, Heegard also made numerous other criticisms of Analysis Situs. Back then math had a lot of back and forth like that. Today, you generally have to be more rigorous to publish, though it depends on the field.

 

[bob012345]

So, the original question asked in this thread is why physicists are so informal with math. Please critique the following argument (and please be kind...). It seems to me if math is just a tool physicists use to represent physical ideas, the rigor of the mathematical argument does not infer the rigor of the physical argument. Proof in the mathematical realm does not infer proof in the physical realm otherwise any proposed theory would automatically imply proof as meaningful physics merely by being rigorously true mathematically. So physicists are more concerned with the physics ideas represented than the mathematical rigor of the presentation.

 

[symbolipoint]

So, the original question asked in this thread is why physicists are so informal with math. Please critique the following argument (and please be kind...). It seems to me if math is just a tool physicists use to represent physical ideas, the rigor of the mathematical argument does not infer the rigor of the physical argument. Proof in the mathematical realm does not infer proof in the physical realm otherwise any proposed theory would automatically imply proof as meaningful physics merely by being rigorously true mathematically. So physicists are more concerned with the physics ideas represented than the mathematical rigor of the presentation.

Wow! That is confusing but I agree with you, what you said.

 

[Hornbein]

The challenge in general relativity wasn't the math, it was in convincing the reader that the math described the real world.

 

[Sargon38]

I realize of course that this will probably not apply to all physicists, but at least every physicist in my university's math department is very unrigorous when it comes to mathematics. This is frustrating because some of the physics material seems genuinely interesting, but the lack of an axiomatic approach, proofs, and rigor makes it incredibly unappealing (I've skimmed the course notes).

I remember (vaguely, it's long ago) to have been frustrated throughout my education about exactly that. It took me a few decades to understand why it was wrong to be frustrated about it, simply because for years, I hadn't understood the difference between mathematics and physics: I thought that physics was a part of mathematics. It isn't.

Physics is a natural science, based upon the fundamental principle of observational science: experiment is the final judge. Mathematics is not: it is a "mind game". Mathematics needs rigor because it doesn't have any other "truth feedback". If you're sloppy with your thinking in maths, you essentially have nothing left. Physics (like any other natural science) always has experiment as a final feedback, so it doesn't need to be as rigorous in its "thinking". The lab is the final judge.
Historically, mathematics hasn't always been as rigorous as today, but they could get away with it because historically, a lot of what people called "mathematics" was IN FACT physics. Most of Euclidean geometry is in fact physics: it's the physics of physical space. That the sum of the angles of a triangle equals 180 degrees can be MEASURED to a certain extend, when you draw a physical triangle on a piece of paper. Euclid's axiomatic system was as leaky as hell, but there wasn't anything wrong in it, because you got physical experimental feedback. Zeno's paradox was evidently seen as wrong because we could experimentally observe that a hare could overtake a turtle.

It is only when mathematics really got abstract, say from the 19th century onward, and lost its evident link with physical reality that rigor was needed because it was the only thing left that could separate maths from bullshit.

Physics can get away with "Euclidean style" sloppyness, because in the end, the "truth feedback" is experiment, not logical rigor. Well, until a few decades ago with physicists delving in string theory and the like :-)

For instance, we don't need existance and uniqueness proofs for Navier-Stokes equations, because we know that water flows. If we have an operational technique, suggested by "sloppy math" that allows us to calculate in a non-rigourous way how water would flow, and we OBSERVE that we're doing pretty well, we have the scientifically sound experimental feedback that makes the falsification test of our operational calculational procedure work.

And in the end, we KNOW that water won't exactly flow as those Navier-Stokes equations describe, because we KNOW that water is made up of molecules, and is not a mathematical "real number" liquid. So in the end, we don't care if the Navier-Stokes equations, which include ERRONEOUS simplifications, have mathematical solutions. If those equations SUGGEST an operational calculational procedure that WORKS OUT pretty well in the lab, that's the best we can hope for. We *know* that if ever those Navier-Stokes equations have exact solutions, they won't be describing water flow perfectly in any case, because they have been derived upon approximations of what water physically is. Water is not made out of infinitesimal mathematical dots, but out of finite-sized molecules.

Physics is a natural science. Mathematics is a thought game. These are two totally different human activities.
Mathematics turns out to be a huge toolbox in which physicists and engineers can delve, but only as a toolbox. It is not a part of their core activity, even though on the surface, they are doing similar things.

It took me decades to understand that.

 

[martinbn]

Your example with Euclidean geometry is really bad. There is nothing slopy or non rigorous about, then or now.

 

[Sargon38]

Your example with Euclidean geometry is really bad. There is nothing slopy or non rigorous about, then or now.

It is generally well-known that the original axiomatic approach of Euclid is simply incomplete and erroneous as to today's standards of mathematical rigor. But there's nothing wrong with that, exactly because the "experimental" return from the drawing indicates that one is not making mistakes.
But even the proof of the very first theorem in the very first book by Euclid, where he proves that on any segment, one can construct an equilateral triangle, is flawed in a known fashion: there's no way to prove that the two circles have a point in common.

Actually, I use Euclid's geometry to show the errors and to make people think about what is a correct proof, and what looks like one but is in fact erroneous.
You can have a look at this site:
https://mathcs.clarku.edu/~djoyce/java/elements/elements.html

The example I talk about can be seen here:

https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI1.html

read the critical analysis.

Euclid's Elements is a great way to learn about mathematical rigour by counter example.

One of the modern attempts of doing it the right way are Hilbert's axioms:

https://en.wikipedia.org/wiki/Hilbert's_axioms

There are many more axioms needed (as Hilbert indicates) than the 5 original Euclidean axioms to make Euclidean geometry rigorous. But as it concerns observable geometry that you can "see with your eyes" on paper, most of these theorems are "evident on the drawing" (which, however, is not part of mathematical rigor).

In fact, the sloppiness of Euclid was one of the reasons, in the 60-ies and 70-ies, to throw it out of HS curricula and to introduce Bourbaki-style geometry (a.k.a. modern mathematics). I think it is more pedagogical to go by standard Euclid and show how it is actually only semi-rigorous.

The site on Clark's University is very enlightening concerning the rights and wrongs of Euclid's Elements. I find this actually very instructive as an introduction to mathematical proof as I said earlier.

One can say that Euclid's Elements is very good physics :-) It fits perfectly in this thread ;-)

 

[AndreasC]

It is generally well-known that the original axiomatic approach of Euclid is simply incomplete and erroneous as to today's standards of mathematical rigor.

It's not up to par with modern standards of rigor of SOME fields, but nevertheless it's pretty solid, much more solid than most of what physicists do... I still think you're kinda stretching your point with axiomatic euclidean geometry (since Elements and its method was basically the blueprint for more rigorous math that came later, even though it wasn't as refined as the modern stuff) but that's some very interesting work on the Elements.

 

[gmax137]

Yesterday's rigor is today's slop. This applies all over the place, not just mathematics. Given time, experts will find fault with everything preceding their own work.

 

[mathwonk]

I also accepted the lack of rigor and completeness in Euclid as generally well known, until I actually read it. Having done so, in my opinion it is extremely rigorous, with one or two easily augmented exceptions.

Namely, essentially the only flaws/omissions I am aware of, are the failure to clarify that a line [and also a circle, Thank you Sargon38] separates the plane into two "sides", and to state, before applying it, the fact that one can "move" triangles isometrically in the plane.

I came to read and appreciate Euclid after encountering the following essay by Hartshorne, who acknowledges the gaps plugged by Hilbert, but also emphasizes the many insights to be gained from the original.
https://citeseerx.ist.psu.edu/docum...&doi=036ef18677fd575f435db78b46ace648f864f253

Note that the biggest mathematical story associated to Euclid's axioms is not the intuitively obvious ones he omitted, but the famous 5th "parallel" postulate that he included, which many people long believed logically unnecessary. When hyperbolic geometry was discovered centuries later, he was proven right.

 

[MidgetDwarf]

I studied pure math at university BS, and in next semester will be my final semester in MS pure math program. Oh i was also 4 classes short from a physics BS while doing my undergrad.

Did not take much applied math courses besides the required cal sequence, ode, and intro linear.

Take physics at what it is, and think of it as applied math .

 

[MidgetDwarf]

I studied pure math at university BS, and in next semester will be my final semester in MS pure math program. Oh i was also 4 classes short from a physics BS while doing my undergrad.

Did not take much applied math courses besides the required cal sequence, ode, and intro linear.

Take physics at what it is, and think of it as applied math .

 

[Sargon38]

It's not up to par with modern standards of rigor of SOME fields, but nevertheless it's pretty solid, much more solid than most of what physicists do... I still think you're kinda stretching your point with axiomatic euclidean geometry (since Elements and its method was basically the blueprint for more rigorous math that came later, even though it wasn't as refined as the modern stuff) but that's some very interesting work on the Elements.

The point I wanted to make is not to say that Euclid is "bad math" but that Euclid is "good physics". And that we should look upon "sloppy math physics with good lab outcomes" the same way. In other words, for me, Euclid is a good illustration why one shouldn't get frustrated at sloppy math in physics, for exactly the same reason that you (and many people since centuries before you) found Euclid actually more than OK.

Why do you even think that Euclid is "pretty solid" ? Because you look at the drawings and you can easily excuse the omissions, because "on the drawing it is obvious". But looking at the drawing is PHYSICS. It is a lab experience in physical space. You "see" that the circles cut, you "see" that a point is on the left side of a line and so on. It is "mathematicians's obsessional nitpicking" to try to prove the "obvious". But it is obvious because you make a physical observation.

If you weren't making a physical drawing in real space, but you were trying to prove Euclid's theorems purely formally by just writing lines of statements and logical schemes to justify the next line, you would see obviously that the theorems are not provable. It is the drawing that "suggests" by physical visual observation that "tells you the obvious", not logical deduction (at all).

Imagine you have in a non-drawing deduction:

blah blah ...

"we have segment [AB]"

blah blah

"we have segment [CD]"

There's no way in which you can now define point E as being the intersection of segments [AB] and [CD]. No logical scheme allows you to conclude from the existence of [AB] and of [CD] that there exists a point E belonging to both.

In the same way, if you have a circle C1, and a cercle C2, there's no logical way in which you can define point E2 as being the/an intersection point of C1 and C2. If you were writing this purely as a proof without drawings, you wouldn't even think of introducing such a point, as there's no logical scheme that allows you to do so.

The strictly only reason why you can get away with tricking people into believing that this point exists, is because you propose a physical observation after an "experiment" on a piece of paper, or a computer screen, and people OBSERVE that there's an intersection point.

Well, that's physics.

There's nothing wrong with that. Nobody will actually DOUBT that on a segment [AB], you can construct an equilateral triangle. It is OBVIOUS from physical observation. It is mathematical nitpicking to want to get this "right", and that's why you think Euclid is "pretty solid". Because it is indeed pretty solid physics. There's even overkill. Because many "proofs" are overkill and the drawings are obvious.

What I wanted to say is that other physics is similar: the fact that it works out when you make observations is the final justification, and you don't need 'mathematical rigor' as mathematicians need it when they are talking about abstractions when there are NO OBSERVATIONS to justify their conclusions.

Personally, I find Euclid extremely enlightening, exactly because it is on this borderline between math and physics. To me, it is the perfect illustration of both sides, and why what is OK in physics, is problematic in mathematics.

Euclid without drawings (without physical observation) fails totally. Euclid with drawings is solid and obvious.

As I said earlier, it took me ages to understand this. I was just as frutrated as the TS when I was younger. I thought that physicists were a bunch of failed mathematicians. I got a stroke when I read up on QFT. It is only many years later that I understood this. I'm writing this because it might shorten the time of frustration of some, like the TS :-)

 

[AndreasC]

Why do you even think that Euclid is "pretty solid" ? Because you look at the drawings and you can easily excuse the omissions, because "on the drawing it is obvious".

But the point of Euclid was the basic postulates, and deriving everything from them. A lot of the stuff in Elements was completely obvious intuitively but he still went through the trouble to prove it. Other ancient geometers also took issue with some of the proofs which they noticed were somewhat wrong, they didn't content themselves with the drawings looking right. What I'm saying is that in general, Euclid sticks to the axiomatic method and he uses it pretty rigorously, except for a few points where he makes a hidden assumption or two, simply because he just kinda didn't realize probably. If he did he would probably have tried to prove or postulate his assumptions. And if that's not what Euclid had in mind, certainly other geometers of his time did think that way because they criticized some of these proofs in exactly the same lines, and tried to prove some of these propositions.

Because many "proofs" are overkill and the drawings are obvious.

But that's the thing, it's not overkill because he was clearly trying to do it rigorously. That he messed up here and there is only understandable, given that it was one of the first attempts to do math like that.

However Euclid was physics in another sense. Geometry and astronomy were pretty much identified back then.

 

[PeroK]

The point I wanted to make is not to say that Euclid is "bad math" but that Euclid is "good physics".

It's true that a physicist won't get far without basic geometry and trigonometry, but that doesn't make these physics. By definition, they are mathematics. You are free to have your own definitions of mathematics and physics, but you should recognise these as such and not imagine that the rest of the academic community is going to adopt your definitions.

 

[Sargon38]

What I'm saying is that in general, Euclid sticks to the axiomatic method and he uses it pretty rigorously, except for a few points where he makes a hidden assumption or two, simply because he just kinda didn't realize probably. If he did he would probably have tried to prove or postulate his assumptions.

But in what way is that different from a physicist who's doing "sloppy math" as compared to a rigorous mathematician ?

Except maybe that you're saying that Euclid was trying to be rigorous, probably thought he was, while the physicist *knows* that he's cutting corners, in order to get things done ?

Now, the question is: suppose that an evil genie would have told Euclid that his first proof already went wrong, but let's assume that Euclid wouldn't know how to solve the issue, would:

1) this mean that Euclid gives up on writing the Elements, because he realises that his full-rigor approach doesn't work and he's out of inspiration, so to hell with it ? If it ain't perfect, don't do it ?

or:
2) this mean that Euclid would still go on writing the Elements, knowing that he cut a corner here and there, but overall, he has a great textbook on geometry and mathematics in general ?

Should the physicist, knowing Haag's theorem, give up on calculating a Feynman diagram, or should he continue doing so, knowing that he's cutting corners ? Should he wait doing a calculation to be tested in the lab until everything is axiomatically correctly demonstrated, and in the case it isn't, should he just stop doing physics, or should he go on ?

 

[Sargon38]

It's true that a physicist won't get far without basic geometry and trigonometry, but that doesn't make these physics. By definition, they are mathematics. You are free to have your own definitions of mathematics and physics, but you should recognise these as such and not imagine that the rest of the academic community is going to adopt your definitions.

I know of course that elementary geometry is thought in mathematics courses. It's tradition.

But it is actually the first physics course if you think about it. Geometry was ENTIRELY inspired by physical space. It is the physical theory of space. It was the basis of a lot of engineering in antiquity. It got essentially all of its inspiration from physical observation and measurement. Before the Greeks formalized it, it was an engineering discipline.
The reason why Euclidean geometry was thought to be the only "possible" one until the discovery of non-Euclidean geometries (which ARE mathematical theories) was because of its intimate relation to physical space as we observe it.

And the exact reason why Euclid could get away with holes in its demonstrations were exactly because of the physical space and the observations therein.

As I said earlier, if geometry were "maths" one shouldn't use drawings in physical space, one should use formal proofs as in, say, linear algebra or in analysis. It should be pure text. There shouldn't be any "lab observations" on drawings in physical space.

In a way, Euclidean geometry (in the Greek sense) stands to physical space, such as Newton's mechanics stands to the motion of solid objects in physical space. Most people consider Newtonian mechanics as physics, even though one can "mathematize" it to a high extend.

Both physical theories have inspired mathematicians to set up mathematical theories by abstraction, generalisation and variation. Modern mathematical geometry and analysis respectively. But at their basis, we have two physical theories: one of physical space, and one of physical motion.

 

[martinbn]

@Sargon38 You keep repeating this, but can give any examples where Euclid was sloppy and not rigorous?

 

[Sargon38]

@Sargon38 You keep repeating this, but can give any examples where Euclid was sloppy and not rigorous?

I gave a well-known example in post 111: the fact that it is assumed without proof (and it CANNOT be proven within Euclid's axiomatic system) that the two circles have at least a point in common in his first proposition in book I.

Mind you, by "lack of rigor" I do not mean "lack of being exhaustive in the explicitness of the argument". I mean "the reasoning is formally failing". It is not that in the first proposition, Euclid neglected showing that the two circles intersect, but that it was a matter of some work to show they do, work he omitted. No. You simply CANNOT show that they intersect in the frame of his system.

But there are many more. Of course, the most obviously failed demonstrations are those where we have triangle congruences: SAS (proposition IV of book I) The same goes for proposition 8 which is SSS.
The demonstration is essentially: "slide the first triangle onto the second, and you'll see for yourself that they coincide".

Now, I'm actually surprised that people are surprised or even sceptical towards Euclid's "sloppiness" as this is a very well known thing.

If you want to know more about it, I could suggest this article for instance:

https://www.michaelbeeson.com/research/papers/euclid2020.pdf

And, again, this is not to play down Euclid. It is to illustrate how physical feedback can allow for mathematical lack of rigor.

 

[PeroK]

As I said earlier, if geometry were "maths" one shouldn't use drawings in physical space, one should use formal proofs as in, say, linear algebra or in analysis. It should be pure text. There shouldn't be any "lab observations" on drawings in physical space.

You can say it as often as you like, but it's only your eccentric opinion. A diagram can represent a mathematical structure - Venn diagrams in set theory, graphs are uniquitous in mathematics, probability trees and other nodal structures, and graph theory itself. The list goes on.

 

[PeroK]

@Sargon38 This is a rigorous, pure mathematics book. With a graph on the cover. QED

 

[Sargon38]

You can say it as often as you like, but it's only your eccentric opinion. A diagram can represent a mathematical structure - Venn diagrams in set theory, graphs are uniquitous in mathematics, probability trees and other nodal structures, and graph theory itself. The list goes on.

This is of course not what I implied: I didn't say that one cannot use "graphical representations of concepts", such as commutativity diagrams or the like. I'm saying that "geometry on paper" is using physically measurable stuff. That's not just "mind maps" of concepts, but it is about the physical space in which we make the drawings.

You see, the Greek attempts at "mathemathifying geometry" were exactly as one does with a physical theory: one tries to put the foundations, the principles, of (an aspect of) physical reality on paper, and one tries to show that with these foundations, one can arrive at falsifable predictions of observable facts. That's exactly what the Greeks did with the EMPIRICAL aspects of space they were aware of by observation.

You see, the difference between a mathematical approach and a "physical" approach is that the mathematical approach starts with basic principles, and EXPLORES where these principles lead you ; while the physical approach starts with to-be-explained facts, and you try to find the principles BEHIND IT. That last thing is exactly what the Greek geometers wanted to do: to find the principles between KNOWN geometry facts.

The difference is this:

1) did Euclid want to find out where his axioms led him ?

or

2) did Euclid want to show that his axioms could provide us with *known geometry knowledge* ?

I'm inclined to think the latter. In other words, Euclid HAD NO DOUBT that you could construct (in physical space) an equilateral triangle on a segment. He knew this to be a fact. He simply hoped he could show that his axioms allowed him to show that he could derive it. But in the ground, he had not the slightest bit of doubt that the physically observable evidence, that with a compass, you can construct an equilateral triangle on any piece of line drawn.

He was trying to find the "founding principles of those known facts of geometry", He wasn't curious about exploring the logical consequences of a set of axioms he thought up. If it were the latter, he wouldn't have committed all those errors.

If you try to find axioms that can explain "known facts of nature" you're essentially constructing a physical theory. If you are exploring the logical consequences of a set of principles, you're doing mathematics.

At no point it would have occured to Euclid to claim that *in his theory* there's no way to prove that you can construct an equilateral triangle on a segment, as would have been the case if this were pure maths. It HAD to be provable to have an equilateral triangle on a segment. Because we KNEW this to be so on a piece of paper.

Now, of course, I agree that it is a thin line between both, and I'm NOT claiming that geometry is physics PER SE. Modern geometry is mathematics, EXACTLY because modern mathematical theories accept the consequences of their principles, for instance, in affine geometry, many things of "Euclidean geometry" are not provable exactly because affine geometry is more universal than Euclidean geometry.

I was just pointing to Euclidean geometry (in the Elements' sense) as a very good illustration of why what is mathematically not rigorous, is perfectly OK for a physical theory. Exactly because in a physical theory, there is experimental feedback (which is what drawings are in the case of Euclidean geometry, seen as a physical theory of space).

 

[Sargon38]

@Sargon38 This is a rigorous, pure mathematics book. With a graph on the cover. QED

View attachment 342264

If this were done the Elements' way, the intermediate value theorem would be "obvious" and not demonstrated.
That's what I wanted to say: in that book, I suppose one is not using diagrams to conclude about elements of proof, as Euclid does.

 

[PeroK]

This is of course not what I implied: I didn't say that one cannot use "graphical representations of concepts"

Yes, you did:

As I said earlier, if geometry were "maths" one shouldn't use drawings in physical space, one should use formal proofs as in, say, linear algebra or in analysis. It should be pure text.

 

[Sargon38]

Yes, you did:

I hope you make the distinction between "graphical representations of concepts" (that is to say, "mind maps", where the drawings are just a notational aid of abstract ideas, which can just as well be described with text), and "drawings in physical space" where the physical attributes of the drawn objects are measurable.

There's a difference, for instance, between a commutative diagram, as a symbolic representation, and drawing a triangle, where the sides have to be straight lines, and where I can measure the angles and the lengths of the sides.

A graphical representation of concepts is convertible in text. A commutative diagram can be converted into text, without loss of contents. A triangle can't be converted into text, without losing "geometrical measurable and observable information".

When I write "the triangle ABC", I cannot measure its sides with a ruler, I cannot measure its inner angles with a protractor, not more than I can measure the voltage over a resistor when I have the diagram of the circuit, and not the circuit itself.
When I draw a triangle, with a ruler, I can measure its sides, I can measure its angles with a protractor. That's like having the electric circuit on the table, not its diagram.

The point I was making, was that in a physical theory, there's no problem in using observational evidence as a substitute for mathematical rigor, while in a mathematical theory, there can in principle not be any observational evidence. I suggested looking upon Euclid's Elements that way, where a lot of observational evidence from drawings was regularly used as a substitute for mathematical proof (which was lacking in principle, not just by succintness), to argue that, as far as we understood Euclid as being a physical theory of physical space, we could accept that, the same way we accept other supposed "mathematical lack of rigor" in the rest of physics.

In a purely mathematical theory, however, there's no possibility of observational evidence, as mathematical theories talk about abstract ideas that do not relate to anything observable. As such, in mathematics, the very idea of observation as a replacement for logical deduction is fundamentally impossible.

Euclid's Elements is not even conceptually possible as a purely mathematical theory, simply because, after removal of all observable facts in drawings, the errors would be so blatant that they wouldn't even have been made in the first place. It is absolutely evident that Euclid was influenced by physical observation of drawings.

Without drawings, the very concept that the two circles would have intersection points would not even occur.

 

[AndreasC]

But in what way is that different from a physicist who's doing "sloppy math" as compared to a rigorous mathematician ?

I guess there are some similarities in that respect but I think it's still not a perfect analogy because Euclid was trying to do something else than what physicists do today. Indeed, the most interesting thing about Elements is probably exactly the method that he laid out: laying out some principles and trying to prove things based only on those principles. He didn't do it perfectly but it was pretty damn solid for when he did it. So solid that scientists and philosophers utilizing a similar methodology would literally call it the "geometric method".

Now, the question is: suppose that an evil genie would have told Euclid that his first proof already went wrong, but let's assume that Euclid wouldn't know how to solve the issue, would:

1) this mean that Euclid gives up on writing the Elements, because he realises that his full-rigor approach doesn't work and he's out of inspiration, so to hell with it ? If it ain't perfect, don't do it ?

I can't possibly know what exactly Euclid would have done, but I believe he would at first try to prove it, and when he failed, he would have just added it as a postulate. But of course that's a hypothetical that is kind of putting the cart before the horse, since the whole concept of rigor comes out of criticizing previous work. Euclid couldn't possibly have simply skipped ahead centuries to modern formal logic or whatever (actually even the concept of, say, deductive logic etc hadn't yet been formulated). But the key novelty (if it was indeed a novelty, I'm not sure about what the historical record says about it) was that Euclid tried to derive everything from a clean set of first principles, without adding extra things. Did this work? Not completely, but that was the idea. Physics generally doesn't do that, even arguments that are, on the surface, very simple, make a whole host of assumptions, hidden or not, that are justified by intuition, experiment, or even convenience.

 

[AndreasC]

When I draw a triangle, with a ruler, I can measure its sides, I can measure its angles with a protractor

You can't, according to the rules laid out by Euclid. The ruler of euclidean geometry is unmarked. The whole point of it is to go beyond these things.

You see, the difference between a mathematical approach and a "physical" approach is that the mathematical approach starts with basic principles, and EXPLORES where these principles lead you ; while the physical approach starts with to-be-explained facts, and you try to find the principles BEHIND IT. That last thing is exactly what the Greek geometers wanted to do: to find the principles between KNOWN geometry facts.

I disagree with this, because I don't believe this is an accurate description of either physics or mathematics. No mathematician gets entirely rid of intuition and just sits there, grinding away random consequences of axioms. And a huge part of mathematics even today is trying to come up with good definitions and good principles which capture the essence of some structure. I don't remember it it was Clausen or Scholze that I saw saying that most of what he does is really just try to come up with good definitions.

Plus, if you consider things historically, you'll see that practically ALL math is initially related to something "observational", be it physics or whatever. As it develops it often abstracts more and more from it, and this can already be seen in Euclid as well, this doesn't make it any special from modern math.

 

[Frabjous]

One can argue that modern mathematical rigor did not start until the nineteenth century. Something went on before this and most modern mathematics is based upon it. It seems silly not to call it mathematics.

 

[depial]

A way that I've often looked at the difference between the rigor of mathematics and the more fast-and-loose approach in physics is with an analogy to literature. There is a whole scale of 'rigor' in the written (and oral) word, from legalese to poetry, with prose sandwiched in between there. Legalese strives to be precise and unambiguous in grammar and word definitions to nail down meaning (and so happens to be almost impenetrable to many laymen in doing so). Meanwhile poetry uses much more relaxed grammatical rules and word definitions, albeit while often still having clear structure, to explore meaning (all with the potential to be fairly opaque to the uninitiated as well). In between these, prose has a wide range of rigor and accessibility, which largely depends on application (e.g. a dry treatise on history or a fantasy novel).

In this light, it sounds a bit odd if a lawyer complains about poetry's flowery impreciseness, and poets won't likely feel any need to abandon their methods in the face of such criticisms (nor should the lawyer abandon their methods in the face of the poet's complaints).

I fell like the salient point of the original post is in reference to teaching, in specific, taking into account the knowledge base of the students taking the mentioned course. Is there is a better (less 'flowery' or more 'advanced') way to present a semester long course in physics to a term of mathematicians at OP's level? I don't know if we can translate poetry into legalese, but we might be able to find some relevant prose.

 

[martinbn]

One can argue that modern mathematical rigor did not start until the nineteenth century. Something went on before this and most modern mathematics is based upon it. It seems silly not to call it mathematics.

I think Euclid would pass 20th century standards of rigor.

 

[Frabjous]

I think Euclid would pass 20th century standards of rigor.

That isn’t the point I was trying to make.

 

[martinbn]

That isn’t the point I was trying to make.

I know. It is something i was saying in addition.

 

[Sargon38]

I guess there are some similarities in that respect but I think it's still not a perfect analogy because Euclid was trying to do something else than what physicists do today. Indeed, the most interesting thing about Elements is probably exactly the method that he laid out: laying out some principles and trying to prove things based only on those principles. He didn't do it perfectly but it was pretty damn solid for when he did it. So solid that scientists and philosophers utilizing a similar methodology would literally call it the "geometric method".

Yes, I agree that Euclid wanted to establish a purely axiomatic system where everything was logically deduced from those axioms. And yes, he's more or less the inventor of that model of thinking, and yes, that's how most of mathematics is done now. But that were *his intentions*.

My point was that the *actual result*, namely the Elements, was something akin of what physicists do when they are "accused of being sloppy". I think indeed that Euclid didn't want to be sloppy. And as I repeated often, I'm not throwing any accusations at him: what he did was spectacular.

But the result of all that was, nevertheless, and contrary to what Euclid probably wanted, a kind of "physical theory", in the sense that Euclid did get some of the fundamental principles of physical space right, and that the rest was filled in by observation of the actual behavior of drawings in physical space.

Even though Euclid wanted to put down a mathematical theory that stood by itself, the result didn't - but was marvelously *useful and working* if one added in those little perky details *from observation*, which didn't follow from his fundamentals. I find that this is exactly what physicists do when they use a theory to actually calculate things: they "cut corners" on the mathematical side because they are guided by observation. Which is *perfectly OK*.

So you can see my point differently if you want:
Euclid's elements was, on the mathematical side, "close but no sigar", but stood out and still stands out magnificently if we look upon it as a physical theory of the geometry of physical space, where now and then, we put in some missing detail from obvious observation. This is why I prefer to look upon it as a first physical theory of space, even though I know that Euclid wanted it to be purely formal.

My idea was that physicists do something similar when they cut corners in mathematics: they are guided by observation, and don't bother about pesky details that are in any case "obvious" by observation, or could at worst make them stop getting results in the first place.

To argue whether in his time, it was mathematics or physics doesn't make much sense, because there was no distinction between both fields back then. Physics was rather called natural philosophy, and notions such as physical space were or seen as philosophical considerations, or mathematics.

But the key novelty (if it was indeed a novelty, I'm not sure about what the historical record says about it) was that Euclid tried to derive everything from a clean set of first principles, without adding extra things. Did this work? Not completely, but that was the idea. Physics generally doesn't do that, even arguments that are, on the surface, very simple, make a whole host of assumptions, hidden or not, that are justified by intuition, experiment, or even convenience.

Yes, I agree of course that Euclid is considered as the inventor of the axiomatic method (even though it was in the air at that time), and that one could say that he's as such, the father of mathematics as we know it today.
However, if you would think of trying to write a physical theory of static physical space, that tries to describe the physics of how space is made, I think you would do something very close to what Euclid did in his Elements.

 

[Sassan]

There is little respect in physics for semantics. Case in point:
A WAVE is defined as "an oscillation (or disturbance) that travels through a MEDIUM, transferring energy". Then the statement is made that "Electromagnetic waves can travel through VACUUM". If you perform semantic substitution (put the definition of "wave" from the first statement into the second statement where "wave" is mentioned) you end up with a self-contradictory statement because vacuum is not a medium.
Some complain that there is no axiomatic exposition of physics. It is worse than that: There is sometimes little LOGICALLY CONSISTENT connection among certain basic statements, as seen in the WAVE example above. It is as though each statement in physics gives itself the right to define words anyway it wishes, independent of how those words have been used in the very same book, article, or lecture . . . or even the same paragraph!

 

[PeroK]

There is little respect in physics for semantics. ... because vacuum is not a medium.

That's sheer semantics!

 

[gmax137]

A WAVE is defined as ...

In physics, "wave" denotes something that obeys a wave equation, such as $\ddot u = c^2 \nabla ^2 u$

 

[Sassan]

Thank you for your reply.

Imagine an equation describing the behavior of an automobile. Then when someone asks, "What is an automobile?", we simply respond "that which obeys the automobile equation". While this answer may be correct, it is neither complete nor enlightening.

Equations are the end-result of physics inquiry. The earlier phases begin with observing, conceptualizing, hypothesizing, quantifying, experimenting, and deducing conclusions from all that, which can be summarized compactly in the form of a formula -- the formula being the "telegraphic" summary of all that came before it.

My original question about semantics in physics was formulated at the conceptual level. This is an important level of inquiry, because if we don't get things right on this level, then the subsequent formulas may look pretty but will be wrong. It was at this level that Einstein re-conceptualized gravity as the curvature of spacetime due to matter, not at the formula level, which came later on.

 

[gmax137]

There is little respect in physics for semantics.

I think you are correct. Physics respects the ability of a model to predict.

 

[Frabjous]

I think of a wave as the process by which knowledge of a disturbance is transmitted to other points in space at a finite propagation speed.

 

[Sassan]

This is a very good definition, but the word "space" is used generically in order not to commit specifically to either "matter" or "vacuum". We know/see how that knowledge of the disturbance travels through matter, but it is not intuitively clear how it travels through vacuum. Perhaps one should give up intuition at this point, just as we are required to give up intuition in understanding phenomena at the quantum level.

 

[Frabjous]

This is a very good definition, but the word "space" is used generically in order not to commit specifically to either "matter" or "vacuum". We know/see how that knowledge of the disturbance travels through matter, but it is not intuitively clear how it travels through vacuum. Perhaps one should give up intuition at this point, just as we are required to give up intuition in understanding phenomena at the quantum level.

Lack of intuition about a mechanism does not invalidate the definition. There are multiple types of waves. There are waves that propagate through a medium and waves that are self-propagating.

 

[Sassan]

Yes, that is exactly my point about semantic consistency. If from the very beginning a wave is defined without reference to a "medium", then later on when it is learned that a wave can travel through vacuum, no inconsistency arises.
In mathematics, we are very careful that one statement follows from the previous statements, or is at least logically consistent with them. This gives rise to a continuity in thinking. Physicists do not seem to be as sensitive to this conceptual continuity. Even definitions become controversial. There are so much discussion on what F=MA "really means".

 

[gmax137]

If from the very beginning a wave is defined without reference to a "medium", then later on when it is learned that a wave can travel through vacuum, no inconsistency arises.

Well for quite a while most (?) physicists believed that EM "waves" necessarily implied a medium, namely the electromagnetic aether. Progress is made when previous definitions (even those centuries or millenia old) are recognized to be artificially limiting thought.

This is admittedly quite different from mathematics. It seems there you start with definitions set in stone, as it were.

 

[Sassan]

Yes, we know that EM waves propagate through the vacuum, but there is so much controversy amongst physicists as to exactly HOW this happens. I have read many accounts of it by folks with physics degrees, but they tend to accuse one another of committing misconceptions, and that only THEY have the correct explanation. This borders on religion: Accepting a statement on FAITH!
My concern is not so much about the body of knowledge we call physics as it is about teaching it to others. It is sometimes taught as a form of religion: "Trust me when I tell you that ....". The only things trustable in physics are the outcomes of experiments, not the interpretations of those outcomes that are unfortunately subject to so much controversy.
In mathematics definitions are not set in stone; they are conditional: "IF you accept definitions A/B/C (and axioms M/N/P), THEN conclusions XYZ follow from them". But if you don't accept those definitions and axioms, then those conclusions don't follow! The perfect model of this is Euclidean Geometry. It got invalidated when the assumptions were challenged in non-Euclidean Geometry. It is like a chess game with very clear rules, and if you follow those rules the game makes sense.
Not so in physics!

 

[gmax137]

My concern is not so much about the body of knowledge we call physics as it is about teaching it to others. It is sometimes taught as a form of religion: "Trust me when I tell you that ....".

I guess we went to different schools.

 

[Sassan]

I am not alone in this.

The following passage is from:
https://www.quantamagazine.org/what...ition to photons — the,that fill all of space
...............................

Mark Van Raamsdonk remembers the beginning of the first class he took on quantum field theory as a Princeton University graduate student. The professor came in, looked out at the students, and asked, “What is a particle?”

“An irreducible representation of the Poincaré group,” a precocious classmate answered.

Taking the apparently correct definition to be general knowledge, the professor skipped any explanation and launched into an inscrutable series of lectures. “That entire semester I didn’t learn a single thing from the course,” said Van Raamsdonk, who’s now a respected theoretical physicist at the University of British Columbia.