Why are Physicists so informal with mathematics?

[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.