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1. • CommentRowNumber1.
   • CommentAuthorEric
   • CommentTimeJul 3rd 2010
   • (edited Jul 3rd 2010)
   • PermaLink
   Author: Eric
   Format: MarkdownItexMotivated by some recent comments from Ian, I've restarted my "Train Readings" again. I stopped for a while because I'm completely exhausted most of the time and the 30 minute train ride provided a necessary 30 minute vegetative state. In particular, I started with this: >[A new description of orthogonal bases](http://arxiv.org/abs/0810.0812) by [[Bob Coecke|Coecke]], Pavlovic, and [[Jamie Vicary|Vicary]]. The advantage of reading this is there is a good chance I will actually understand it :) I like this and can't help relate it to [[ericforgy:Discrete differential geometry on causal graphs|things I'm interested in]], but I'll try to resist for now. I think it is marginally interesting that "every commutative $\dagger$-Frobenius monoid in FdHilb determines an orthogonal basis." How about the experts in the room? Is this an interesting result? Is this new? To me, it says that a choice of comultiplication amounts to a choice of a basis, which seems fairly obvious. What happens to comultiplication under a change of basis? Let $$\delta |\phi_i\rangle = |\phi_i\rangle \otimes |\phi_i\rangle$$ and $$|\alpha\rangle = \sum_i |\phi_i\rangle\langle \phi_i|\alpha\rangle = \sum_i \alpha_i |\phi_i\rangle$$ then (assuming $\delta$ is linear) $$\delta \alpha = \sum_i \alpha_i |\phi_i\rangle \otimes |\phi_i\rangle.$$ Now change basis with a new basis dependent comultiplication $$\delta'|\phi_i'\rangle = |\phi_i'\rangle \otimes |\phi_i'\rangle$$ we get $$\delta' \alpha = \sum_i \alpha_i' |\phi_i'\rangle \otimes |\phi_i'\rangle.$$ Have we gained anything by doing this? Basis dependent maps seem unnatural in my opinion. I like the idea of categorical quantum mechanics, but I'm not yet convinced that this is the right approach. There must be a more natural way to proceed. Any ideas? Personally, I would be tempted to try to interpret the basis vectors $|\phi_i\rangle$ as objects and $$|\phi_i\rangle\otimes |\phi_j\rangle$$ as a morphism $|\phi_i\rangle\to|\phi_j\rangle$. In this way $$\delta |\phi_i\rangle = |\phi_i\rangle\otimes |\phi_i\rangle$$ might be an [[identity-assigning morphism]]. So maybe what we really want is a category [[internal to]] FdHilb. Or something... I'm just thinking out loud... Ok. I really can't resist one more note... As far as maps $H\to H^2$, my favorite map is the coboundary $$d|\phi_i\rangle = \sum_j \left(|\phi_j\rangle\otimes |\phi_i\rangle - |\phi_i\rangle\otimes |\phi_j\rangle \right).$$ Note $$m\circ d = 0.$$ More generally, I think $d$ is related the kernel of $m$. This has got to be relevant here.

   Motivated by some recent comments from Ian, I’ve restarted my “Train Readings” again. I stopped for a while because I’m completely exhausted most of the time and the 30 minute train ride provided a necessary 30 minute vegetative state.

   In particular, I started with this:

   A new description of orthogonal bases

   by Coecke, Pavlovic, and Vicary. The advantage of reading this is there is a good chance I will actually understand it :)

   I like this and can’t help relate it to things I’m interested in, but I’ll try to resist for now.

   I think it is marginally interesting that “every commutative $\dagger$-Frobenius monoid in FdHilb determines an orthogonal basis.” How about the experts in the room? Is this an interesting result? Is this new? To me, it says that a choice of comultiplication amounts to a choice of a basis, which seems fairly obvious.

   What happens to comultiplication under a change of basis?

   Let

   $$\delta |\phi_i\rangle = |\phi_i\rangle \otimes |\phi_i\rangle$$

   and

   $$|\alpha\rangle = \sum_i |\phi_i\rangle\langle \phi_i|\alpha\rangle = \sum_i \alpha_i |\phi_i\rangle$$

   then (assuming $\delta$ is linear)

   $$\delta \alpha = \sum_i \alpha_i |\phi_i\rangle \otimes |\phi_i\rangle.$$

   Now change basis with a new basis dependent comultiplication

   $$\delta'|\phi_i'\rangle = |\phi_i'\rangle \otimes |\phi_i'\rangle$$

   we get

   $$\delta' \alpha = \sum_i \alpha_i' |\phi_i'\rangle \otimes |\phi_i'\rangle.$$

   Have we gained anything by doing this? Basis dependent maps seem unnatural in my opinion.

   I like the idea of categorical quantum mechanics, but I’m not yet convinced that this is the right approach. There must be a more natural way to proceed.

   Any ideas?

   Personally, I would be tempted to try to interpret the basis vectors $|\phi_i\rangle$ as objects and

   $$|\phi_i\rangle\otimes |\phi_j\rangle$$

   as a morphism $|\phi_i\rangle\to|\phi_j\rangle$. In this way

   $$\delta |\phi_i\rangle = |\phi_i\rangle\otimes |\phi_i\rangle$$

   might be an identity-assigning morphism.

   So maybe what we really want is a category internal to FdHilb. Or something…

   I’m just thinking out loud…

   Ok. I really can’t resist one more note…

   As far as maps $H\to H^2$, my favorite map is the coboundary

   $$d|\phi_i\rangle = \sum_j \left(|\phi_j\rangle\otimes |\phi_i\rangle - |\phi_i\rangle\otimes |\phi_j\rangle \right).$$

   Note

   $$m\circ d = 0.$$

   More generally, I think $d$ is related the kernel of $m$. This has got to be relevant here.

2. • CommentRowNumber2.
   • CommentAuthorEric
   • CommentTimeJul 3rd 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItexQuestion: Given a 0-category $B$ of basis vectors, is it possible to think of a vector as a functor $F:B\to C$ where $C$ is the category of 1-dimensional vector spaces? Oh! Or maybe as a colimit of this functor? I'm not sure how best to say this... I'm thinking that each object $b_i$ gets sent to a 1-dimensional Hilbert space (or something) with $F$ assigning the components, e.g. $$F(b_i) = F_i e_i$$ where $F_i$ is a scalar and $e_i$ is unit basis vector. The colimit would glob these together into a vector. Or something... Or maybe put a different way... Given an object $V$ in $Vect$, if we form the [[category of elements]], is there a way to reconstruct $V$ from its elements? Note: This is an example where if I knew how to precisely formulate my question, it would mean I understood things well enough that I wouldn't need to ask my question, so please read this with a high degree of flexibility in interpretation of what I'm trying to say. Note: At [[category of elements]] I claimed the elements could be repackaged, but I'm not 100% sure how to do it.

   Question:

   Given a 0-category $B$ of basis vectors, is it possible to think of a vector as a functor $F:B\to C$ where $C$ is the category of 1-dimensional vector spaces? Oh! Or maybe as a colimit of this functor?

   I’m not sure how best to say this…

   I’m thinking that each object $b_i$ gets sent to a 1-dimensional Hilbert space (or something) with $F$ assigning the components, e.g.

   $$F(b_i) = F_i e_i$$

   where $F_i$ is a scalar and $e_i$ is unit basis vector. The colimit would glob these together into a vector. Or something…

   Or maybe put a different way…

   Given an object $V$ in $Vect$, if we form the category of elements, is there a way to reconstruct $V$ from its elements?

   Note: This is an example where if I knew how to precisely formulate my question, it would mean I understood things well enough that I wouldn’t need to ask my question, so please read this with a high degree of flexibility in interpretation of what I’m trying to say.

   Note: At category of elements I claimed the elements could be repackaged, but I’m not 100% sure how to do it.

3. • CommentRowNumber3.
   • CommentAuthorIan_Durham
   • CommentTimeJul 3rd 2010
   • (edited Jul 3rd 2010)
   • PermaLink
   Author: Ian_Durham
   Format: MarkdownItex<blockquote><p>I like the idea of categorical quantum mechanics, but I'm not yet convinced that this is the right approach. There must be a more natural way to proceed.</p></blockquote> Well, personally I think the greatest "failing" of QM (and, mind you, it is the most experimentally accurate theory ever devised) is it's treatment of time as absolute. Rovelli attempted to amend that (Relational QM), but there are some problems with his argument. Nevertheless, it is common to compare the QM notion of a basis with the relativistic notion of a frame. Since at its core, physics is nothing more than processes, category theory in its purest form seems perfect to describe all of it, QM or otherwise. But then again, I'm a reductionist.

   I like the idea of categorical quantum mechanics, but I’m not yet convinced that this is the right approach. There must be a more natural way to proceed.

   Well, personally I think the greatest “failing” of QM (and, mind you, it is the most experimentally accurate theory ever devised) is it’s treatment of time as absolute. Rovelli attempted to amend that (Relational QM), but there are some problems with his argument. Nevertheless, it is common to compare the QM notion of a basis with the relativistic notion of a frame.

   Since at its core, physics is nothing more than processes, category theory in its purest form seems perfect to describe all of it, QM or otherwise. But then again, I’m a reductionist.

4. • CommentRowNumber4.
   • CommentAuthorEric
   • CommentTimeJul 3rd 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItexFor the sake of argument, let's assume we are not interest in nature, but a mathematical model of nature. This model, quantum mechanics, is not perfect by any means, but it has a working mathematical foundation. This foundation is generally not presented from the nPOV. For the purposes of this thread, I hope to think about the model of quantum mechanics (not necessarily a model of nature) and try to formulate it from the nPOV in the most natural way possible. I think that is what Coecke is also interested in doing. Specifically, pick up any undergraduate text on quantum mechanics. I'd like to rewrite that book from the nPOV (or cPOV) without getting bogged down by philosophical questions. I'm not trying to come up with a new model of nature. I'm more modestly interested in trying to rewrite the semantics/language of the existing theory. This seems like a worthy effort that might be of some interest to this group. For example, [Kindergarten Quantum Mechanics](http://arxiv.org/abs/quant-ph/0510032) seems to be an effort in a similar direction, but this presentation doesn't "ring true" to me, so I'm interested in finding an alternative presentation.

   For the sake of argument, let’s assume we are not interest in nature, but a mathematical model of nature. This model, quantum mechanics, is not perfect by any means, but it has a working mathematical foundation. This foundation is generally not presented from the nPOV.

   For the purposes of this thread, I hope to think about the model of quantum mechanics (not necessarily a model of nature) and try to formulate it from the nPOV in the most natural way possible.

   I think that is what Coecke is also interested in doing.

   Specifically, pick up any undergraduate text on quantum mechanics. I’d like to rewrite that book from the nPOV (or cPOV) without getting bogged down by philosophical questions. I’m not trying to come up with a new model of nature. I’m more modestly interested in trying to rewrite the semantics/language of the existing theory. This seems like a worthy effort that might be of some interest to this group.

   For example,

   Kindergarten Quantum Mechanics

   seems to be an effort in a similar direction, but this presentation doesn’t “ring true” to me, so I’m interested in finding an alternative presentation.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJul 3rd 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItex@Ian: My understanding is that "classical" QM treats time as absolute, but that QFT is fully special-relativistic and treats time as a class of directions in Minkowski space, just like any other relativistic theory. One could make the analogy you mentioned between bases in QM and frames in SR, but I think it's just an analogy; the problems of QM are not about time per se.

   @Ian: My understanding is that “classical” QM treats time as absolute, but that QFT is fully special-relativistic and treats time as a class of directions in Minkowski space, just like any other relativistic theory. One could make the analogy you mentioned between bases in QM and frames in SR, but I think it’s just an analogy; the problems of QM are not about time per se.

6. • CommentRowNumber6.
   • CommentAuthorIan_Durham
   • CommentTimeJul 3rd 2010
   • PermaLink
   Author: Ian_Durham
   Format: Html@Mike: you are absolutely correct about that. I think, for whatever reason, many (but not all) quantum foundations people tend not to think in field theoretic terms and so we tend never to consider QFT. Oddly, Rovelli, who takes this temporal approach in <a href="http://arxiv.org/abs/quant-ph/9609002">his paper on relational quantum mechanics</a>, also writes about field theory in other papers. I like <a href="http://arxiv.org/abs/1004.3564">Isham's approach</a>, which is a bit different from Coecke's in that it aims for a more unified approach. One of the points Isham makes is about the use of probabilities, though, and I'm not entirely sure I buy this (<a href="http://arxiv.org/abs/1003.5209">QBism</a> is another interesting approach that tackles the probability question head-on). But, like Eric, I would really love to start from the ground up. However, I'd be interested in trying it for physics in general from an empirical standpoint. To me, basic category theory seems to fit this so well. But I want something both powerful and useful in the end and I worry that the formalism and language of more advanced category theory could render it too abstruse. In other words, the effort either has to offer new insights or radically change the way we think about physics for it to be anything other than a mental exercise. That said, I do think it is worth it. So, when I teach introductory physics, I usually teach the class for science but non-physics majors (we have three levels of intro.). Over the course of the past decade I've developed a sort of philosophy of physics largely influenced by the writing of Tom Moore and Dan Schroeder (some of you may know Schroeder from his book on QFT with Michael Peskin). So when I introduce it on the first day of class to these kids, this is roughly what I tell them: The universe is made up of lots of particles. Those particles interact with one another (and exchange yet more particles, i.e. gluons, vector bosons, etc. which I liken to mailmen delivering messages). If all those particles <em>didn't</em> interact, the universe would be a boring place. So, ultimately, we are most interested in the <em>interactions</em> (or lack thereof) between these particles. Now, to understand why I think category theory would be ideal for this, I offer this quote from Awodey's book: "It's the arrows that really matter!" (By arrows, he means morphisms.) In any case, that's how I start out in the first couple of minutes of my class (sans category theory). Everything in the course is built on those basic principles and I reinforce them in every upper level class I teach (I introduce them to the physics majors in our Modern Physics class since they have someone else for intro. usually).

   @Mike: you are absolutely correct about that. I think, for whatever reason, many (but not all) quantum foundations people tend not to think in field theoretic terms and so we tend never to consider QFT. Oddly, Rovelli, who takes this temporal approach in his paper on relational quantum mechanics, also writes about field theory in other papers.
   
   I like Isham's approach, which is a bit different from Coecke's in that it aims for a more unified approach. One of the points Isham makes is about the use of probabilities, though, and I'm not entirely sure I buy this (QBism is another interesting approach that tackles the probability question head-on).
   
   But, like Eric, I would really love to start from the ground up. However, I'd be interested in trying it for physics in general from an empirical standpoint. To me, basic category theory seems to fit this so well. But I want something both powerful and useful in the end and I worry that the formalism and language of more advanced category theory could render it too abstruse. In other words, the effort either has to offer new insights or radically change the way we think about physics for it to be anything other than a mental exercise. That said, I do think it is worth it.
   
   So, when I teach introductory physics, I usually teach the class for science but non-physics majors (we have three levels of intro.). Over the course of the past decade I've developed a sort of philosophy of physics largely influenced by the writing of Tom Moore and Dan Schroeder (some of you may know Schroeder from his book on QFT with Michael Peskin). So when I introduce it on the first day of class to these kids, this is roughly what I tell them:
   
   The universe is made up of lots of particles. Those particles interact with one another (and exchange yet more particles, i.e. gluons, vector bosons, etc. which I liken to mailmen delivering messages). If all those particles didn't interact, the universe would be a boring place. So, ultimately, we are most interested in the interactions (or lack thereof) between these particles. Now, to understand why I think category theory would be ideal for this, I offer this quote from Awodey's book: "It's the arrows that really matter!" (By arrows, he means morphisms.)
   
   In any case, that's how I start out in the first couple of minutes of my class (sans category theory). Everything in the course is built on those basic principles and I reinforce them in every upper level class I teach (I introduce them to the physics majors in our Modern Physics class since they have someone else for intro. usually).
7. • CommentRowNumber7.
   • CommentAuthorEric
   • CommentTimeJul 4th 2010
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   Author: Eric
   Format: MarkdownItexMaybe we could choose a common reference and try to re-express it in category theoretic terms. Any suggestions? The Wikipedia page seems like a reasonable place to start: >[Mathematical formulations of quantum mechanics](http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics) But, not totally unexpected, the Wikipedia article seems to pull us toward [POVM](http://en.wikipedia.org/wiki/POVM) and [PVM](http://en.wikipedia.org/wiki/Projection-valued_measure), which comes full circle to [[quantum operations]].

   Maybe we could choose a common reference and try to re-express it in category theoretic terms. Any suggestions?

   The Wikipedia page seems like a reasonable place to start:

   Mathematical formulations of quantum mechanics

   But, not totally unexpected, the Wikipedia article seems to pull us toward POVM and PVM, which comes full circle to quantum operations.

8. • CommentRowNumber8.
   • CommentAuthorIan_Durham
   • CommentTimeJul 4th 2010
   • (edited Jul 4th 2010)
   • PermaLink
   Author: Ian_Durham
   Format: Html<blockquote><p>But, not totally unexpected, the Wikipedia article seems to pull us toward POVM and PVM, which comes full circle to quantum operations.</p></blockquote> Right! Though POVMs are considered passé by some (not really sure why). At any rate, that's a really good idea - start with the Wikipedia page and see where it leads us. (Note: I'll be away tomorrow.) As a minor side note, this got me thinking about super-simplified applications of category theory and it occurred to me that one way to use it to represent Newton's third law, for instance, is to say that Newton's third law in category-theoretic terms could be that the category of all forces (as represented by some kind of morphism) must necessarily have an opposite category (or something like that). I have some similar ideas for Newton's other two laws, but I don't think it's really all that relevant. But it was an interesting "warm up" for my brain. Edit: I am thinking outloud above so don't be too harsh.

   But, not totally unexpected, the Wikipedia article seems to pull us toward POVM and PVM, which comes full circle to quantum operations.

   
   Right! Though POVMs are considered passé by some (not really sure why). At any rate, that's a really good idea - start with the Wikipedia page and see where it leads us. (Note: I'll be away tomorrow.)
   
   As a minor side note, this got me thinking about super-simplified applications of category theory and it occurred to me that one way to use it to represent Newton's third law, for instance, is to say that Newton's third law in category-theoretic terms could be that the category of all forces (as represented by some kind of morphism) must necessarily have an opposite category (or something like that). I have some similar ideas for Newton's other two laws, but I don't think it's really all that relevant. But it was an interesting "warm up" for my brain.
   
   Edit: I am thinking outloud above so don't be too harsh.
9. • CommentRowNumber9.
   • CommentAuthorEric
   • CommentTimeJul 4th 2010
   • (edited Jul 4th 2010)
   • PermaLink
   Author: Eric
   Format: MarkdownItexCaleb O'Loan's PhD thesis also looks like a great place to start: [Topics in Estimation of Quantum Channels](http://arxiv.org/abs/1001.3971) His Chapter 1 looks nicely paced. Rewriting classical mechanics would also be fun, and I'm sure is possible, but people are less familiar with thinking of classical mechanics in terms of states and processes. I'd also propose treating time as discrete so we can avoid "infinitesimal morphisms" for the time being. For example, objects could be QM systems at given times and a morphism takes a QM system at time $t$ to time $t+\Delta t$ for some finite $\Delta t$. This can be made rigorous by integrating the infinitesimal morphisms and will make life easier. PS: I've now created [[ericforgy:Categorical Quantum Mechanics]]

   Caleb O’Loan’s PhD thesis also looks like a great place to start:

   Topics in Estimation of Quantum Channels

   His Chapter 1 looks nicely paced.

   Rewriting classical mechanics would also be fun, and I’m sure is possible, but people are less familiar with thinking of classical mechanics in terms of states and processes.

   I’d also propose treating time as discrete so we can avoid “infinitesimal morphisms” for the time being. For example, objects could be QM systems at given times and a morphism takes a QM system at time $t$ to time $t+\Delta t$ for some finite $\Delta t$. This can be made rigorous by integrating the infinitesimal morphisms and will make life easier.

   PS: I’ve now created Categorical Quantum Mechanics (ericforgy)

10. • CommentRowNumber10.
    • CommentAuthorIan_Durham
    • CommentTimeJul 4th 2010
    • (edited Jul 4th 2010)
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    Author: Ian_Durham
    Format: HtmlOK, cool. I'll be away today (my wife is going to kill me for even turning on my computer), but I'll start thinking about this stuff and will hopefully have something to add tomorrow when I get back. <blockquote><p>Rewriting classical mechanics would also be fun, and I'm sure is possible, but people are less familiar with thinking of classical mechanics in terms of states and processes.</p></blockquote> Right, but I would love to get it so that there was just <em>one</em> way that students thought about physics. It would be so much easier to get them thinking about states and processes, for example, right off the bat. But starting with QM is fine by me. I can always do the other stuff on my own and see what happens.

    OK, cool. I'll be away today (my wife is going to kill me for even turning on my computer), but I'll start thinking about this stuff and will hopefully have something to add tomorrow when I get back.
    

    Rewriting classical mechanics would also be fun, and I'm sure is possible, but people are less familiar with thinking of classical mechanics in terms of states and processes.

    
    Right, but I would love to get it so that there was just one way that students thought about physics. It would be so much easier to get them thinking about states and processes, for example, right off the bat. But starting with QM is fine by me. I can always do the other stuff on my own and see what happens.
11. • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 4th 2010
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    Author: Todd_Trimble
    Format: MarkdownItex> Right, but I would love to get it so that there was just *one* way that students thought about physics. Really? Feynman and Wheeler spent their entire careers encouraging the opposite attitude. Let a thousand flowers bloom! ;-)

    Right, but I would love to get it so that there was just one way that students thought about physics.

    Really? Feynman and Wheeler spent their entire careers encouraging the opposite attitude. Let a thousand flowers bloom! ;-)

12. • CommentRowNumber12.
    • CommentAuthorIan_Durham
    • CommentTimeJul 5th 2010
    • PermaLink
    Author: Ian_Durham
    Format: Html<blockquote><p>Really? Feynman and Wheeler spent their entire careers encouraging the opposite attitude. Let a thousand flowers bloom! ;-)</p></blockquote> While I am certainly not even remotely close to as smart as either of them, I really think that that attitude is partly why we have yet to achieve a unified theory. I wrote a whole <a href="http://arxiv.org/abs/1001.4304">essay on it</a> recently (that happened to be one of the few things I've written that other people actually liked). Aside from that, though, it would also be a lot better from a teaching standpoint.

    Really? Feynman and Wheeler spent their entire careers encouraging the opposite attitude. Let a thousand flowers bloom! ;-)

    
    While I am certainly not even remotely close to as smart as either of them, I really think that that attitude is partly why we have yet to achieve a unified theory. I wrote a whole essay on it recently (that happened to be one of the few things I've written that other people actually liked). Aside from that, though, it would also be a lot better from a teaching standpoint.
13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2010
    • PermaLink
    Author: Urs
    Format: MarkdownItex> that attitude is partly why we have yet to achieve a unified theory. No, Ian, that makes no sense.

    that attitude is partly why we have yet to achieve a unified theory.

    No, Ian, that makes no sense.

14. • CommentRowNumber14.
    • CommentAuthorIan_Durham
    • CommentTimeJul 5th 2010
    • PermaLink
    Author: Ian_Durham
    Format: Html<blockquote><p>No, Ian, that makes no sense.</p></blockquote> Well, it made sense to the committee that judged the latest FQXi essay contest since that essay was awarded third place. So maybe it makes no sense to you, but it made sense to someone other than me.

    No, Ian, that makes no sense.

    
    Well, it made sense to the committee that judged the latest FQXi essay contest since that essay was awarded third place. So maybe it makes no sense to you, but it made sense to someone other than me.
15. • CommentRowNumber15.
    • CommentAuthorIan_Durham
    • CommentTimeJul 5th 2010
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    Author: Ian_Durham
    Format: HtmlHere's an imperfect analogy, but one that I think gets the basic idea across that I'm talking about. I emphasize that I think this is only <em>part</em> of the problem. Obviously the whole problem is much deeper than this, but it's a place to start (IMHO). So imagine we have a book (let's call it "The Universe") in some alien language that we've only partly translated. But we've translated a portion of it (e.g. GR) into Dutch and a portion (e.g. QM) into German. Now, Dutch and German are both germanic languages and have many similarities, but they're still not the same language. It would be better to have this book translated into a single language so that the some of the subtleties aren't lost between chapters.

    Here's an imperfect analogy, but one that I think gets the basic idea across that I'm talking about. I emphasize that I think this is only part of the problem. Obviously the whole problem is much deeper than this, but it's a place to start (IMHO).
    
    So imagine we have a book (let's call it "The Universe") in some alien language that we've only partly translated. But we've translated a portion of it (e.g. GR) into Dutch and a portion (e.g. QM) into German. Now, Dutch and German are both germanic languages and have many similarities, but they're still not the same language. It would be better to have this book translated into a single language so that the some of the subtleties aren't lost between chapters.
16. • CommentRowNumber16.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 5th 2010
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    Author: Todd_Trimble
    Format: MarkdownItexIt might be better to let Feynman speak for himself: Terry Tao posted a link to lectures given by Feynman <a href="http://terrytao.wordpress.com/2009/07/15/feynmans-lectures-online/">here</a>. Highly recommended. Professor Tao comments: > Of particular interest to mathematicians is his second lecture "The relation of mathematics and physics". He draws several important contrasts between the reasoning of physics and the axiomatic reasoning of formal, settled mathematics, of the type found in textbooks; but it is quite striking to me that the reasoning of unsettled mathematics - recent fields in which the precise axioms and theoretical framework has not yet been fully formalised and standardised - matches Feynman's description of physical reasoning in many ways. I suspect that Feynman's impressions of mathematics as performed by mathematicians in 1964 may differ a little from the way mathematics is performed today. To me, Feynman forcefully argued how the basic facts of physics can be arrived at via multitudinous approaches, and that axiomatic frameworks in physics are quite relative and fluid. Therefore a dogmatic attitude on the 'right' approach is generally to be shunned. Compare Gian-Carlo Rota's remarks on the phenomenology of mathematics in his Indiscrete Thoughts (which are primary? theorems or definitions?). Wheeler in his autobiography describes a big sea change in his thinking over the course of his career, roughly from an "all is particle" way of thinking to an "all is wave" way of thinking. Of course that is a gross oversimplification; what really comes across is the fluidity and flexibility of his thinking, away from the ossification which can result from too heavy dependence on axiomatized approaches.

    It might be better to let Feynman speak for himself: Terry Tao posted a link to lectures given by Feynman here. Highly recommended. Professor Tao comments:

    Of particular interest to mathematicians is his second lecture “The relation of mathematics and physics”. He draws several important contrasts between the reasoning of physics and the axiomatic reasoning of formal, settled mathematics, of the type found in textbooks; but it is quite striking to me that the reasoning of unsettled mathematics - recent fields in which the precise axioms and theoretical framework has not yet been fully formalised and standardised - matches Feynman’s description of physical reasoning in many ways. I suspect that Feynman’s impressions of mathematics as performed by mathematicians in 1964 may differ a little from the way mathematics is performed today.

    To me, Feynman forcefully argued how the basic facts of physics can be arrived at via multitudinous approaches, and that axiomatic frameworks in physics are quite relative and fluid. Therefore a dogmatic attitude on the ’right’ approach is generally to be shunned. Compare Gian-Carlo Rota’s remarks on the phenomenology of mathematics in his Indiscrete Thoughts (which are primary? theorems or definitions?).

    Wheeler in his autobiography describes a big sea change in his thinking over the course of his career, roughly from an “all is particle” way of thinking to an “all is wave” way of thinking. Of course that is a gross oversimplification; what really comes across is the fluidity and flexibility of his thinking, away from the ossification which can result from too heavy dependence on axiomatized approaches.

17. • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2010
    • PermaLink
    Author: Urs
    Format: MarkdownItexJust one remark on this: > Feynman forcefully argued how the basic facts of physics can be arrived at via multitudinous approaches, and that axiomatic frameworks in physics are quite relative and fluid. It's of course entirely possible to have multitudinous approaches _and_ each of them fully axiomatized. And preferably a bunch of theorems relating these axiomatizations. We see this in those parts of physics that have been pretty much settled: there are several ways to approach and to formalize classical mechanics. And each of them as well as their relationship is pretty well understood mathematically. I think the main issue is that of the large multitude of vague approaches in physics, a smaller multitude -- but still a multitude -- will eventually survive and admit formalization. The danger is to pick an appproach for formalization that is not yet ready for it, or to axiomatize the wrong aspect. There was recently (in terms of years) an interesting incident illustrating this process, where somebody proposed an axiomatization of one of the hottest topics in present fundamental physics research and encountered the criticism that this axiomatization -- while certainly axiomaizing _something_ -- did not in fact capture what physicists meant to capture and found of interest to capture.

    Just one remark on this:

    Feynman forcefully argued how the basic facts of physics can be arrived at via multitudinous approaches, and that axiomatic frameworks in physics are quite relative and fluid.

    It’s of course entirely possible to have multitudinous approaches and each of them fully axiomatized. And preferably a bunch of theorems relating these axiomatizations.

    We see this in those parts of physics that have been pretty much settled: there are several ways to approach and to formalize classical mechanics. And each of them as well as their relationship is pretty well understood mathematically.

    I think the main issue is that of the large multitude of vague approaches in physics, a smaller multitude – but still a multitude – will eventually survive and admit formalization. The danger is to pick an appproach for formalization that is not yet ready for it, or to axiomatize the wrong aspect.

    There was recently (in terms of years) an interesting incident illustrating this process, where somebody proposed an axiomatization of one of the hottest topics in present fundamental physics research and encountered the criticism that this axiomatization – while certainly axiomaizing something – did not in fact capture what physicists meant to capture and found of interest to capture.

18. • CommentRowNumber18.
    • CommentAuthorIan_Durham
    • CommentTimeJul 5th 2010
    • PermaLink
    Author: Ian_Durham
    Format: Html<blockquote><p>Therefore a dogmatic attitude on the 'right' approach is generally to be shunned. </p></blockquote> I agree that it's a fine line to straddle. In no way do I advocate a 'right' approach in the sense of favoring one formalism over another, but the fact remains that the "language" of QM (not including QFT) and GR are very different. One approach to unification has been the field theoretic approach that seems to be favored on this site and there's nothing wrong with it. But let me quote from a book by Mike Fortun and Herb Bernstein called <em>Muddling Through: Pursuing Science and Truths in the 21st Cnetury</em>: <blockquote><p>Language matters. Language is essential to reason, and can’t be gotten rid of so easily with a few new machines. Somewhere along the line - no matter how long that line is - every experiment, every mathematical equation, every pure numerical value will have to find its way into words.</p></blockquote> I just happen to think we'll get a lot further if we are all "speaking the same language."

    Therefore a dogmatic attitude on the 'right' approach is generally to be shunned.

    
    I agree that it's a fine line to straddle. In no way do I advocate a 'right' approach in the sense of favoring one formalism over another, but the fact remains that the "language" of QM (not including QFT) and GR are very different. One approach to unification has been the field theoretic approach that seems to be favored on this site and there's nothing wrong with it.
    
    But let me quote from a book by Mike Fortun and Herb Bernstein called Muddling Through: Pursuing Science and Truths in the 21st Cnetury:
    

    Language matters. Language is essential to reason, and can’t be gotten rid of so easily with a few new machines. Somewhere along the line - no matter how long that line is - every experiment, every mathematical equation, every pure numerical value will have to find its way into words.

    
    I just happen to think we'll get a lot further if we are all "speaking the same language."
19. • CommentRowNumber19.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 5th 2010
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexThere is no contesting that language matters -- a great deal. > I just happen to think we'll get a lot further if we are all "speaking the same language." I think it's important to be conversant in a number of languages!

    There is no contesting that language matters – a great deal.

    I just happen to think we’ll get a lot further if we are all “speaking the same language.”

    I think it’s important to be conversant in a number of languages!

20. • CommentRowNumber20.
    • CommentAuthorIan_Durham
    • CommentTimeJul 5th 2010
    • PermaLink
    Author: Ian_Durham
    Format: Html<blockquote><p>I think it's important to be conversant in a number of languages!</p></blockquote> As do I, but I find it easier if the two (or more) people in a conversation are speaking the same language at the same time.

    I think it's important to be conversant in a number of languages!

    
    As do I, but I find it easier if the two (or more) people in a conversation are speaking the same language at the same time.