Why are Physicists so informal with mathematics?

[AndreasC]

They are according to Terence Tao. See the link above. Tao directly contradicts what you say.

But neither my statement nor I believe Tao's is universal here... Obviously some things are intuitive even without having to go through this process. It's why physicists can get away with not knowing the rigorous foundations and still extract good results.

I don't believe this is the way mathematicians have worked in the past 150 years. The onus is on you to prove that this is true. In particular, this statement is entirely false.

I have a few arguments to that effect. First, the cutoff is not as sharp or as old as you claim. As I said before, even as soon as Poincare things are not that rigorous. Second, look at the importance conjectures have. A conjecture is not proven. However you have things such as the Langlands program and many others which are entirely about exploring the ramifications of conjectures, or trying to prove them. But how were the conjectures formulated in the first place? They did not follow from rigorous foundations directly, or they would be proved. But they are not just random assertions either, they are somehow special, and seem "likely" to be true. In a sense, you could say that the whole process until a conjecture is proved is the moment of intuitive discovery, before it is polished and made rigorous, suspended in time for years.

Now I don't have to work much to prove that most mathematics used in physics was either not developed rigorously or could have developed some other way. Most of it developed before the 20th century, so that's that... At the boundaries of course, the situation changes and you are right.

 

[PeroK]

Second, look at the importance conjectures have. A conjecture is not proven.

An unproven conjecture is not non-rigorous mathematics. Those are two very different things.

 

[PeroK]

But how were the conjectures formulated in the first place?

Through rigorous mathematics.

They did not follow from rigorous foundations directly

Yes they did.

But they are not just random assertions either, they are somehow special, and seem "likely" to be true. In a sense, you could say that the whole process until a conjecture is proved is the moment of intuitive discovery, before it is polished and made rigorous, suspended in time for years.

This is a fantasy.

 

[TurtleKrampus]

Well, to put it more simply, quantum field theory uses a bunch of operators on a Hilbert space to describe "fields", such as the electromagnetic field etc, which give rise to "particles". In essence all these are partial differential equations, whose solutions are promoted to linear operators, via "quantization". That's all fine when the underlying PDE is simple, as is the case with the Klein-Gordon or Dirac equations. These can generally be solved, and the appropriate Hilbert space to study their "quantized" version is called a Fock space. We generally know how their solutions (called free fields) act on the Fock space.

Unfortunately, when you want to have fields interacting with each other, you end up with PDEs involving multiple different functions coupled in convoluted ways. Just Google "standard model Lagrangian". Almost every different letter you see is a different field. Now apply on that the Euler-Lagrange equations and you get an absolutely insane system of horribly coupled PDEs. Of course nobody with the whole thing at once, we just look at parts of it. One such part are the famous Yang-Mills equations: https://en.m.wikipedia.org/wiki/Yang–Mills_equations

The existence problem for the YM equations is a Millennium Prize problem. Actually, it's not just that one, none of the other realistic interacting QFTs have been solved in 4 spacetime dimensions. The reason the other ones aren't a Millennium Prize problem is probably mostly that many people don't think they even really have a solution, for various reasons, never mind they are still used.

But to quantize, say, the Klein-Gordon equation, the best way is to just solve the ("classical") PDE first, and then to promote the solution to an operator in a specific sense. So how are we supposed to quantize the interacting ones when we can't solve them? Physicists have some workarounds. Probably the most commonly used one is a type of functional integral called a path integral. This is another tool that came out of physics (originally to describe Brownian motion if I'm not mistaken) that has been applied to other areas of math. The idea is, you somehow integrate on a space of functions. Of course, to integrate you need a measure. For some specific functional integrals, this measure is known. For the path integrals of QFT, there is no rigorous formalization of the measure as of yet. Nevertheless, it is used.

And then on top of all of that, you do perturbation theory. What's that? Well, we have a Hilbert space we don't really know, and operators representing fields that we have quantized, nevermind the fact that we don't rigorously know how they act on that Hilbert space (or even if we can rigorously consider these actions, because they relate to PDEs we don't know the solutions to), and we want to approximate the solutions to various problems regarding their action on these spaces, using power series. Great.

Perhaps you will find it amusing to learn that these power series have divergent terms. But maybe you already heard that, and heard that you can just do renormalization, etc. Indeed, renormalization generally fixes the problem, and Epstein-Glaser theory shows how you do that rigorously, starting from first principles, in a manner that is not ad hoc. Only physicists usually don't do that and follow a much less rigorous counterterm procedure, that is easier to work with. But at least we know we can cure the divergences. Trouble is, even AFTER you cure these divergences in the terms of the series, the series STILL diverges if you include every term, as in, it has ZERO radius of convergence. The physicist answer to this? "Well I'll just keep the first few terms of the series, which don't diverge". Well, in some cases people use some other summation schemes, like Borel summation etc. But sometimes that doesn't work.

So, to summarize, we start from PDEs that we don't know how to solve or if they even have solutions, we quantize them via integration measures that don't exist, and then we approximate the solutions to various problems using series that don't converge, by just ignoring the rest of the series, at least when we can get each term to converge. And it's not even a cutting edge theory, it's been around for decades. Not just that, but it is probably THE most successful physical theory ever, that has yielded the most precise predictions. This is how physicists learn to be less formal with math.

To learn about QFT, you may be interested in these books, written mostly for mathematicians, by mathematicians:

https://www.amazon.com/dp/0821847058/?tag=pfamazon01-20
https://www.amazon.com/dp/1316510271/?tag=pfamazon01-20

The second one is essentially a more digested version of the first one, including only the things that for the most part are known, but being significantly bigger. It's also interesting to see Talagrand's comments throughout the text indicating his struggle to understand why various things work. Really that's the main strength of the book imo, the fact that when he covers something that is very suspicious but nevertheless works, he says it explicitly. However I'm not sure how much you would get out of these books without further background into physics. Maybe you could try reading the Arnold book I mentioned, then maybe something like Quantum Theory for Mathematicians by Brian Hall, and then the Talagrand book (or the Folland book if you prefer). You will also see how much of QM and QFT really is just representation theory, and see why it was a huge motivator for its development.

I should've responded to this sooner, but this message is too large for me to possibly respond to its entirety (mostly because when I've tried I get bored less than midway). Anyway, thanks for your messages, I've read them I just can't really respond to them if that makes sense. I'm not really interested in self studying QFT, at least at the moment. I'm using most of my "study time" for either my URS or just regular classes. Once again, I really appreciate the answers.

 

[TurtleKrampus]

What I said was this, which you reacted to:


That was the best I could think at the current time. As I plainly said, the comment is not perfect. Have you a thought along the lines of the original posted topic about Mathematics differently handled between Physicists and Mathematicians, and if you want to share with readers here, then say those thoughts.

Yeah, I don't have / didn't want to think of a better analogy, sorry if this sounds harsh. I don't care how physicists do math, not even most math done by mathematicians is thought of rigorously (though there are plenty of cases of just bashing, which you may usually start with rigor so you don't waste time). You have intuition, and see where that leads you.

Again, my problem was in the exposure, please disregard the original post I guess.
Just the other day I had a really awful experience with my Physics professor while introducing preliminary knowledge to Lagrangian mechanics. The basic gist is since the professor basically never mentions the type of domain / codomain of functions (implicitly or explicitly), him writing f(X), where X is a function of a real variable t, as opposed to say something like $(f \circ X)$ lead to a lot of unnecessary ambiguity, that did indeed lead to unnecessary confusion, since at a later point we mention the derivative of f and write f(X(t)) as opposed to f(X)(t).
There are some important details that I also left out, but I don't really want to write out since they'd require a lot more context and I honestly cba to write all of it out.

 

[AndreasC]

Through rigorous mathematics.

Let me put it another way. Most mathematicians believe the Riemann hypothesis is correct. But this suspicion is not grounded in some kind of rigorous proof, obviously. It's just a suspicion, supported by non rigorous arguments. Nevertheless, this suspicion is part of the process of real math even if it is not officially admitted as theorem. In fact, the conjecture would not have been formulated and widely investigated like that had there not been an intuitive jump from what was rigorously known to something that still to this day isn't. It's also telling that entire fields are founded on conjectures. If mathematicians didn't admit non-rigorous methods at all, these fields would he considered a waste of time, until the conjectures were proven or disproven.

One example that I just thought of is Perelman's proof of the Poincare conjecture. Perelman is of course thought of as the one who proved it, but his proof was more like a proof sketch. It still had some very non trivial gaps, non-rigorous "jumps" that he made to reach the solution, that were later filled in by other mathematicians (I believe it was Cao and someone else? I don't remember, you can look it up). This to me reveals a that the creative jump is usually one that is non-rigorous, and supported by (informed) intuition.

Looking back at your original post, I realize that I actually don't really disagree with it much, and perhaps I didn't express myself well enough either. The main point of disagreement is that I don't think most of the unrigorous math in physics is really a "simplification" of a rigorous result, for instance the way calculus is used in physics with hand wavy infinitesimals etc is close to how calculus was originally conceived, before being formalized and made rigorous with epsilons and deltas. But other than that, I don't really disagree that rigor was immensely helpful in building the complex mathematical structures we have today.

 

[TurtleKrampus]

Let me put it another way. Most mathematicians believe the Riemann hypothesis is correct. But this suspicion is not grounded in some kind of rigorous proof, obviously. It's just a suspicion, supported by non rigorous arguments. Nevertheless, this suspicion is part of the process of real math even if it is not officially admitted as theorem. In fact, the conjecture would not have been formulated and widely investigated like that had there not been an intuitive jump from what was rigorously known to something that still to this day isn't. It's also telling that entire fields are founded on conjectures. If mathematicians didn't admit non-rigorous methods at all, these fields would he considered a waste of time, until the conjectures were proven or disproven.

One example that I just thought of is Perelman's proof of the Poincare conjecture. Perelman is of course thought of as the one who proved it, but his proof was more like a proof sketch. It still had some very non trivial gaps, non-rigorous "jumps" that he made to reach the solution, that were later filled in by other mathematicians (I believe it was Cao and someone else? I don't remember, you can look it up). This to me reveals a that the creative jump is usually one that is non-rigorous, and supported by (informed) intuition.

Looking back at your original post, I realize that I actually don't really disagree with it much, and perhaps I didn't express myself well enough either. The main point of disagreement is that I don't think most of the unrigorous math in physics is really a "simplification" of a rigorous result, for instance the way calculus is used in physics with hand wavy infinitesimals etc is close to how calculus was originally conceived, before being formalized and made rigorous with epsilons and deltas. But other than that, I don't really disagree that rigor was immensely helpful in building the complex mathematical structures we have today.

The suspicion that the Riemann hypothesis is true is mostly due to how it relates to the structure of prime numbers, it being false would produce very interesting mathematics. Do not be deceived, there are results in both ways, there are results that hold if the Riemann hypothesis is true and those that hold if it's false.
Regardless, mathematics isn't necessarily the study of proving theorems, conditional results (i.e. if A then B) are still considered mathematics, and completely rigorous. (A premise doesn't have to be true for what's stated to be rigorous / valid.
There are also multiple mathematicians that believe / want to believe that the Riemann hypothesis is false, while they are a minority they do exist.

Humans are flawed creatures, researches when writing papers won't write every single mathematical step, they're bound to mistakes.
Most of them are somewhat naive, you recognize the similarity between things and think "since this holds for X, this is bound to be true here because I can turn this into X", when there's a subtle error in the thinking. Just the other day I thought "surely the set of affine transformations on X forms a ring, since I can embed them in a matrix algebra", this argument was flawed because the embedding doesn't preserve addition in 1 entry (mind you, I should've known better, as if this were true it'd be a very well known result).
People who do mathematics, specially with very long arguments, don't have time to check the validity of every small proposition they "swear" is true, specially when at a sufficiently large level you can't simply google "is X true?" and get an answer.

Does this mean that mathematics is unrigorous? I believe not at all, because first and foremost the published material will always have formal language / very easily formalized language, there should never be any ambiguity about what is meant. Secondly, the peer review process exists for this very reason, people will read what you write, and they'll spot those very propositions and check if they do indeed hold.

I don't care about how people arrive at the idea, I only care about its exposition. How people think of anything is their business, and I don't care at all.

But like, calculus HAS been formalized. There's no need for 'new' objects, which I honestly believe will only give a false sense of understanding and that do not help at all with computation.

 

[AndreasC]

The suspicion that the Riemann hypothesis is true is mostly due to how it relates to the structure of prime numbers, it being false would produce very interesting mathematics.

Right, that's my point, as with many other conjectures, there are arguments, that are not rigorous but nevertheless sound plausible and indicate research directions etc.

My general point is that there is a point during discovery or during devising a proof where mathematicians have to take a leap, that they typically fill up later. They didn't always fill it up later, but today the structures they deal with are incredibly complex and it has proven necessary to do that.

But like, calculus HAS been formalized. There's no need for 'new' objects, which I honestly believe will only give a false sense of understanding and that do not help at all with computation.

Right, calculus has been formalized. But now we get to physics and how physicists use math. The thing is, physicists are first of all constrained and informed by physical reality, on top of the mathematical structure. For instance, for a body thrown upwards in a gravitational field, you can tell its trajectory will have a critical point, without doing any math. You can also tell it will be continuous, etc. So while in this particular example you could be very formal with it, it's kind of a burden to carry all that baggage. You won't encounter the pathological cases that mathematicians deal with so you can safely just ignore complications to keep things easy.

That works most of the time, but not all of the time. Sometimes you need some more serious math to really get to the bottom of something. But that's only a small part of physics research, so it's not what most physicists are taught, although maybe they should.

 

[martinbn]

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

 

[AndreasC]

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

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

 

[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 ?