Feynman's status as a mathematician

[gamow99]

How does Feynman rank as a mathematician in comparison to pure mathematicians of the 20th century, we'll forget about Euler and Gauss. I'm more interested in logic than I am math and consequently I was not able to get beyond calculus, linear algebra and differential equations and some set theory, so I'm really not capable of evaluating Feynman's math work. I am quite skeptical of pure mathematics, especially set theory. I find it suspicious and unrigorous. Feynman on the other hand applied mathematics to physics and consequently I am much more willing to believe his mathematics than say Wiles' proof of Fermat's Last Theorem.

 

[jedishrfu]

I think Feynman was an excellent applied mathematician but his focus and passion was physics and bongo drums among other things.

Here's a video featuring Feynman talking about Physicists and Mathematicians:



which may shed some light on how he viewed himself as a Physicist first and foremost.

With respect to your post, I am always saddened when I hear that someone is skeptical of something after admitting they have yet to understand it. To me its a way of rationalizing why I can't learn something new because its just not important or just not useful.

Math in all its forms is quite beautiful and elegant but as you dig deeper into it you will find you can easily lose that sense of wonder. One of my favorite books that to me brings back this wonder is the Math 1001 book by Elwes. There are many pure math topics that you can read and ponder and wonder how did someone come up with that and how could I use it or why would I even want to use it? and that is the wonder...

 

[SteamKing]

I am quite skeptical of pure mathematics, especially set theory. I find it suspicious and unrigorous.

This is quite a surprising view to take.

Mathematics has been focused on rigor, especially since the early 19th century. For about the first 150 years, the calculus developed more or less by the seat of the pants, and it wasn't until Cauchy published his Cours d'Analyse in 1821 that concepts like the limit were established in a rigorous manner. (Cauchy is the one who devised the epsilon-delta proof for limits).

http://en.wikipedia.org/wiki/Augustin-Louis_Cauchy

Cauchy's contemporary Gauss had burnished his reputation by establishing number theory on a theorem-proof basis.

You mentioned Wiles' proof of Fermat's Last Theorem. The length of this proof is quite extraordinary; the published form of the proof is more than 100 pages, and several mathematicians spent years checking it (and found a few errors which Wiles was able to correct). IDK how you could say that such a great effort was 'unrigorous'.

 

[pwsnafu]

I am quite skeptical of pure mathematics, especially set theory. I find it suspicious and unrigorous.

This statement makes no sense, the entire point of set theory (and especially axiomatic set theory) is to establish rigor in mathematics.

Feynman on the other hand applied mathematics to physics

What does that have to do with rigor?

 

[gamow99]

That's why I told you to try the references.

This is quite a surprising view to take.

Mathematics has been focused on rigor, especially since the early 19th century. For about the first 150 years, the calculus developed more or less by the seat of the pants, and it wasn't until Cauchy published his Cours d'Analyse in 1821 that concepts like the limit were established in a rigorous manner. (Cauchy is the one who devised the epsilon-delta proof for limits).

http://en.wikipedia.org/wiki/Augustin-Louis_Cauchy

Cauchy's contemporary Gauss had burnished his reputation by establishing number theory on a theorem-proof basis.

You mentioned Wiles' proof of Fermat's Last Theorem. The length of this proof is quite extraordinary; the published form of the proof is more than 100 pages, and several mathematicians spent years checking it (and found a few errors which Wiles was able to correct). IDK how you could say that such a great effort was 'unrigorous'.

Let me give you a few reasons. Let's first consider the distinction between math and logic. The only distinction I can see is that math tends to focus on terms whose definitions for which there is a high degree of consensus. With some authors it is not obvious if they are working on logic or on math. Godels theorems is one good example. There is very little consensus in higher logic and I have strong reason to believe that virtually all modal logic is wrong. So at that boundary where the distinction between math and logic is blurry there is a high likelihood that what we believe now will not be believed in 100 years.

 

[gamow99]

the entire point of set theory (and especially axiomatic set theory) is to establish rigor in mathematics.

Is this what you're trying to argue?

1. The whole point of set theory is to establish rigor.
2. Therefore set theory has been successful in establishing rigor.



Is there a decision procedure in set theory? No, there isn't.

 

[Mark44]

... I'm more interested in logic than I am math and consequently I was not able to get beyond calculus, linear algebra and differential equations and some set theory, so I'm really not capable of evaluating Feynman's math work. I am quite skeptical of pure mathematics, especially set theory.

How can you be skeptical of something that you know so little about? There's a lot more to mathematics than calculus, linear algebra, and differential equations, courses that are typically covered in the first year or two of college.

I find it suspicious and unrigorous. Feynman on the other hand applied mathematics to physics and consequently I am much more willing to believe his mathematics than say Wiles' proof of Fermat's Last Theorem.

"Believing" mathematics is an unfortunate turn of phrase, IMO. It's more important to understand it.

Let me give you a few reasons. Let's first consider the distinction between math and logic. The only distinction I can see is that math tends to focus on terms whose definitions for which there is a high degree of consensus.

Can you give a few examples of logic terms for which there is no agreement on their meaning?

With some authors it is not obvious if they are working on logic or on math. Godels theorems is one good example. There is very little consensus in higher logic and I have strong reason to believe that virtually all modal logic is wrong. So at that boundary where the distinction between math and logic is blurry there is a high likelihood that what we believe now will not be believed in 100 years.

 

[homeomorphic]

Although I am in almost complete ignorance of Wiles proof, I think it is highly unlikely that mathematicians would not spot errors in the solution of such a high profile problem. There would be great rewards for spotting any errors. Undoubtedly, whoever did so successfully would get media coverage. I don't necessarily take Fermat's last theorem as a 100% true fact without the slightest grain of doubt, not having checked it myself, but I would be quite surprised if it was wrong. So, I consider it almost surely true, and if I had to place any bets, I would always bet on it being true.

On the other hand, there are some reasons to be suspicious of pure math, and, indeed, the extreme complexity and length of many of the proofs are part of the reason to be suspicious. If this sounds like a bunch of hot air, consider the classification of finite simple groups. It was considered a theorem as far back as the 1980s. However, the proof wasn't fully completed until 2004. Even now, I'm not completely sure that every last detail has been checked, but it seems more secure, since people have worked on simplifying the proof and cutting down its extreme length.

http://en.wikipedia.org/wiki/Classification_of_finite_simple_groups

As far as Feynman goes, his physics had some impact on math, which I don't understand all that well. I studied lattice TQFT, which draws on his path integrals. I don't know how to rank him as a mathematician, though. For that matter, it's really hard to rank anyone as a mathematician.

Cauchy seems to have gotten close to the epsilon-delta stuff (in some cases, all you had to do is formalize his definitions, I believe), but I think that was really Weierstrass's doing, a bit later on.

 

[gamow99]

Can you give a few examples of logic terms for which there is no agreement on their meaning?

Concept, instance, particular, valid, logical consequence, necessity, possibility, impossibility, only, means, tautology, consistent, contradictory, can, must, believe, know, all, some, true.

Let me give a few examples:

A contradiction has the form A and not A, well, if that's true is: 'Some men go to the beach and some men do not go to the beach', a contradiction?'

Let's take 'all'. All the coins in my pocket are quarters. All hydrogen have one proton. Are these the same all? The first all is accidental and the second all is necessary. This distinction has been barely noticed by logicians.

Let's take 'possible'. It is usually defined as 'x is possible if it x exists in one possible world.' That's obviously circular.

Let's take logical consequence. It is commonly defined with truth tables that you are probably familiar with but given that definition of logical consequence 'if the sky is blue then Schrödinger's equation is true' is a true statement and it is quite obvious that the first does not entail the second.

 

[homeomorphic]

"Believing" mathematics is an unfortunate turn of phrase, IMO. It's more important to understand it.

Unfortunately, math these days proves a lot of things that aren't really understandable by any mere mortal, such as the 4-color theorem or the classification of finite simple groups. Check-able, but not understandable (of course, maybe that's what you meant by understanding). In fact, from my experience, I got the impression that, on the whole, people in the math community care more about producing these check-able, but non-understandable facts than they do about producing understanding of the type that's relatively easy to come by in undergraduate math and with more difficulty in graduate level math, where you have a gut feeling for why things work. People think of rigor as being really formal and writing everything down precisely, but that doesn't guarantee that there are no mistakes. It's more rigorous to have an intuitive idea of why it works as well as a formal proof because you have an extra check that it actually makes sense.

After you are done formally checking a proof, step by step, there's a leap of faith that you have to take in order to believe the theorem that doesn't quite occur when you have a deep understanding of how it works. So, in a sense, I would say that's believing mathematics. No one understands the 4-color theorem or the classification of finite simple groups, and I'm guessing, Fermat's last theorem, too. They just believe it because they've checked it. I'm not saying that the 4-color theorem isn't true. I'm sure it's true. It's just that no one understands why it's true. They just checked why, and in that case, they had a computer do part of the checking. It's true that the people who did the proof came to an understanding that it is true because they checked it, though. If you go through all the checking work, you can have that if you like, although in some cases, it is probably a lot more trouble than it's worth.

 

[Mark44]

Concept, instance, particular, valid, logical consequence, necessity, possibility, impossibility, only, means, tautology, consistent, contradictory, can, must, believe, know, all, some, true.

Let me give a few examples:

A contradiction has the form A and not A, well, if that's true is: 'Some men go to the beach and some men do not go to the beach', a contradiction?'

No. In the second clause there's an implied "other" in there, as in "some other men don't go to the beach."

Let's take 'all'. All the coins in my pocket are quarters. All hydrogen have one proton. Are these the same all?

"All" has the same meaning in both cases, "every coin in your pocket" and "every hydrogen atom".

The first all is accidental and the second all is necessary. This distinction has been barely noticed by logicians.

Let's take 'possible'. It is usually defined as 'x is possible if it x exists in one possible world.' That's obviously circular.

In what world is that how "possible" is usually defined? A more usual definition is that an event is possible if there is a nonzero probability of the even occurring. I have never seen it defined in terms of "possible worlds."

Let's take logical consequence. It is commonly defined with truth tables that you are probably familiar with but given that definition of logical consequence 'if the sky is blue then Schrödinger's equation is true' is a true statement and it is quite obvious that the first does not entail the second.

And so what? All that has to happen in a logical implication is that the conclusion part be true when the hypothesis part is true. By definition, implications are also true when the hypothesis is false, independent of whether the conclusion is true. What's your point?

 

[SteamKing]

A contradiction has the form A and not A, well, if that's true is: 'Some men go to the beach and some men do not go to the beach', a contradiction?'

It's a logical contradiction only if you show that the same set of 'some men' going to the beach is the same set of 'some men' who do not go. You are trying to convert the indefinite 'some' into a specific description, which just ain't so.

 

[gamow99]

It's a logical contradiction only if you show that the same set of 'some men' going to the beach is the same set of 'some men' who do not go. You are trying to convert the indefinite 'some' into a specific description, which just ain't so.

That is correct but now you have a new problem on your hands: what is indefinite and what is definite and how do you write an algorithm which will distinguish all definite cases from all indefinite cases. You have plugged one whole of the dyke but a new one has emerged.

 

[gamow99]

No. In the second clause there's an implied "other" in there, as in "some other men don't go to the beach."
"All" has the same meaning in both cases, "every coin in your pocket" and "every hydrogen atom".

If all hydrogen have one proton, then if I find a hydrogen then it will have one proton.
If all the coins in my pocket are quarters, then the next coin that enters my pocket will be a quarter which is clearly false. So something is wrong here.

In what world is that how "possible" is usually defined? A more usual definition is that an event is possible if there is a nonzero probability of the even occurring. I have never seen it defined in terms of "possible worlds."

Have you read a text on modal logic?

And so what? All that has to happen in a logical implication is that the conclusion part be true when the hypothesis part is true. By definition, implications are also true when the hypothesis is false, independent of whether the conclusion is true. What's your point?

So if I tell you that I have found a proof for Fermat's Last Theorem and that my proof is the sky is blue does that mean I have found a proof of FLT?

 

[pwsnafu]

Is there a decision procedure in set theory? No, there isn't.

Decision procedure for what? And what does that have to do with rigor?

 

[SteamKing]

That is correct but now you have a new problem on your hands: what is indefinite and what is definite and how do you write an algorithm which will distinguish all definite cases from all indefinite cases. You have plugged one whole of the dyke but a new one has emerged.

It's 'hole' in the dike, not 'whole' in the dyke, which could have a whole 'nother meaning entirely.

But that's the point. Making general observations about which men go to the beach and which men do not is not intended to be intellectually rigorous.

Let's parse this statement:

"If all the coins in my pocket are quarters, then the next coin that enters my pocket will be a quarter which is clearly false. So something is wrong here."

Yes, there is. You have made the unwarranted assumption that all of the coins in your pocket now must be the same type as any other coin which you may add to your pocket at a later time. You are attempting to treat ordinary language, as imprecise as it is, as if the meaning of a statement which is made can mean anything you want it to mean. By doing so, it becomes impossible to communicate any thoughts between two people.

 

[gamow99]

Let's parse this statement:

"If all the coins in my pocket are quarters, then the next coin that enters my pocket will be a quarter which is clearly false. So something is wrong here."

Yes, there is. You have made the unwarranted assumption that all of the coins in your pocket now must be the same type as any other coin which you may add to your pocket at a later time. You are attempting to treat ordinary language, as imprecise as it is, as if the meaning of a statement which is made can mean anything you want it to mean. By doing so, it becomes impossible to communicate any thoughts between two people.

Let's try to determine what 'all' means:

All x have the relation R to y iff if anything is x then it has the relation R to y.

All hydrogen have one proton iff if anything is hydrogen then it has one proton.
All triangles have three sides iff if anything is a triangle then it has three sides.

Let's now see if this works for the future.
If all hydrogen have one proton then if anything is hydrogen in the future then it is has one proton.
Is that correct? It seems so.
If anything is a triangle in the future then it is has one proton.
Is that correct? It seems so.

Let's now take the following:
All US presidents are male.
Does that work for the future?
If anything is a US president in the future, then it is male.
Does that work? No, it doesn't.
It's the accidental 'all', not the necessary 'all'.

What that all means is the following:
All x have the relation R to y iff if anything was x in the past then x has the relation R to y and if anything is x in the future then it is possible that it has the relation R to y and it is possible that it does not have the relation R to y.

 

[jbriggs444]

The ideal mathematical realm is timeless. Sets do not change. Objects do not change. Relations do not change. Either A is a subset of B or it is not. There is no notion of "it was a subset then, but is not now".

You can, nonetheless, use this timeless realm to model time. Instead of modelling "the coins in my pocket" as set P, model it as a set-valued function of time, P(t).

There is no contradiction between "when t=1, for all x in P(t), x is a quarter" and saying "when t=2, for some x in P(t), x is not a quarter". Putting a dime in your pocket at time 1.5 does not change the value of P(1).

 

[jedishrfu]

The original purpose of this thread was to discuss Feynman as a mathematician and now we digress into whether mathematics is a valid topic for study.

I suggest the thread be closed.

 

[Orodruin]

Closed for moderation.