EDIT: I’ve edited my long post above. Fixed typo’s and clarified technical points I made.
[quote]BlakedaMan wrote:
A lot of people think that linear algebra is dumb because nth dimensions are meaningless, as are complex eigenvectors and such, but they’re actually essential in removing limitations of natural numbers that cause incongruities in not only math, but in physical systems that are described by mathematical expressions.[/quote]
It’s also impossible to do quantum physics without linear algebra.
[quote]BlakedaMan wrote:
stokedporcupine8 wrote:
duffy,
I apologize for making that so long. Hopefully you find it engaging enough to wade through.
This is in response to your post on peano axioms, I just didn’t want to quote the entire thing.
It is important to have formalized integer descriptions, but most of those axioms are common sense arithmetic. Their real value is in showing us the limitations of mathematics as it stands. In reality though, a lot of these issues are solved by complex numbers. A lot of people think that linear algebra is dumb because nth dimensions are meaningless, as are complex eigenvectors and such, but they’re actually essential in removing limitations of natural numbers that cause incongruities in not only math, but in physical systems that are described by mathematical expressions.[/quote]
You don’t quite understand the point of axioms. First, if the point of writing down axioms was indeed to “(show) us the limitations of mathematics as it stands”, then there is no point in writing down axioms, since just how limited some set of axioms is depends on how you write the axioms down. This was part of my whole point: write the peano axioms (formally) one way, and you get some result, write them (formally) another way, you get another. In the end both ways of writing the formal axioms (well, there’s more then two) depend on their informal versions.
In reality, the point of writing down axioms is to capture concrete examples of what your trying to study. Of course, one can do this by writing down the axioms either informally or formally–that is, either in a language lacking a formal semantics or one with a formal semantics. Either way, once you have written down a set of axioms you have in essence captured the objects you wish to study–they are the objects that satisfy the primitive terms of the axioms. So, when someone asks a mathematician, “just what are numbers”, the standard answer is, “they are anything that satisfies the axioms that codify the things I study”. This of course isn’t a definition of numbers–they have no such thing–but it does give meaning to what the mathematician says when she states something like "All natural numbers … ".
Hence, without some sort of axioms your subject matter remains in limbo. For example, an astute high school student may ask their teacher, “Just why is it the case that 2+2=4”, or maybe ask “Just why is it the case that such and such rules for adding fractions hold”. The correct answer, the only answer that can be given, is that those things are so because they are true of the objects we wish to study in the parts of mathematics we call arithmetic. We know that that is true because we have some axioms that specific just what the objects of study in arithmetic look like (The Peano axioms).
As for your point about complex numbers, they–or, rather imaginary quadratic number fields in general–help us prove certain facts about integers, for sure. That does not mean though that those same questions aren’t “resolved” by the axioms. This is a distinction I was hoping not to get into–and avoided in my last long post–but there is a difference between entailment–the idea that all models that satisfy one statement satisfy another statement–and provability–the idea that from some fixed set of axioms and some rules of inference one can generate another statement. While in the last analysis (no pun intended) one may not be able to prove certain facts about integers without recourse to more sophisticated facts about imaginary quadratic number fields, it still is the case that, say, the second-order Peano axioms entail all those facts. Now you may ask what good entailment is, but that is a long discussion in itself. Suffice to say that entail is not only a theoretically important notion but one with much practical important in various situations.
Another good point to bring up is that those quadratic number fields that you bring up have their own set of axioms! Linking back up to my original point, if it weren’t for ring/field axioms and the associated definitions for quadratic number fields, the entire method of complex analysis wouldn’t be available to use in the first place. This theory–these axioms and definitions–are needed to get the subject matter off the ground. So, even if one cannot prove certain facts about integers which generally require complex analysis from the Peano axioms alone, one can certainly prove them from the Peano axioms plus the ring/field axioms plus the definitions of quadratic number fields.
The point is that however you cut it, there’s no escaping axioms in mathematics–even when you think they aren’t at play, they are. They certainly are needed for more than just proving the limitations of our theories.
[quote]Bill Roberts wrote:
BlakedaMan wrote:
A lot of people think that linear algebra is dumb because nth dimensions are meaningless, as are complex eigenvectors and such, but they’re actually essential in removing limitations of natural numbers that cause incongruities in not only math, but in physical systems that are described by mathematical expressions.
It’s also impossible to do quantum physics without linear algebra.
[/quote]
ah, there’s the group theory approach. SU2 to the max!
It’s been so many years that I had to do it that I can’t now do a darn thing in it, but there are all kinds of useful things to calculate or derive for which I still think it is correct to say that you need linear algebra.
Certainly in all kinds of things involved in physical chemistry.
[quote]Bill Roberts wrote:
Stokedporcupine, thank you for that explanation.
I always had a suspicion that Hofstadter had done a little too much LDS (as Kirk would say) but never could figure what the fundamental problem was with his rather acid-tripped book.[/quote]
I can’t say much about Hofstadter because I never read the book. Two quick points: most people who know of Godel’s incompleteness theorem seem to have as much understanding of it as Godel did in 1931, which is a shame because mathematics has come far since then in the relevant areas. The second point is that the sorts of self referential statements that Godel uses in parts of his proof are not essential to proving the theorem. Of course, they’re essential for proving it his way, but his way is monstrous at best. Don’t get me wrong, Godel’s approach is insightful in many respects, and Godel himself was surely a genius of the first order. It is amazing that someone proved incompleteness before the rise of model theory at all.
So I suppose I can summarize thusly: the study of self reference and the possibility of defining truth for a language in that language are extremely important and interesting topics, but at heart they are not essential to the idea of incompleteness. Also, the application of Godel’s theorem–and thus the application of it’s implications for self reference and truth definitions–are limited to certain sorts of formal languages–mainly, the one’s where compactness holds.
[quote]Bill Roberts wrote:
It’s been so many years that I had to do it that I can’t now do a darn thing in it, but there are all kinds of useful things to calculate or derive for which I still think it is correct to say that you need linear algebra.
Certainly in all kinds of things involved in physical chemistry.[/quote]
Maybe, I’m not an expert. There’s also a close connection between the algebra and the group. Perhaps an expert can weigh in.
I think I do actually understand the points of axioms and I wasn’t arguing with you at all, I was simply making the point that the specific set of axioms does in fact show the limits of real integers. Your last paragraph even agrees with that point.
[quote]stokedporcupine8 wrote:
BlakedaMan wrote:
stokedporcupine8 wrote:
duffy,
I apologize for making that so long. Hopefully you find it engaging enough to wade through.
This is in response to your post on peano axioms, I just didn’t want to quote the entire thing.
It is important to have formalized integer descriptions, but most of those axioms are common sense arithmetic. Their real value is in showing us the limitations of mathematics as it stands. In reality though, a lot of these issues are solved by complex numbers. A lot of people think that linear algebra is dumb because nth dimensions are meaningless, as are complex eigenvectors and such, but they’re actually essential in removing limitations of natural numbers that cause incongruities in not only math, but in physical systems that are described by mathematical expressions.
You don’t quite understand the point of axioms. First, if the point of writing down axioms was indeed to “(show) us the limitations of mathematics as it stands”, then there is no point in writing down axioms, since just how limited some set of axioms is depends on how you write the axioms down. This was part of my whole point: write the peano axioms (formally) one way, and you get some result, write them (formally) another way, you get another. In the end both ways of writing the formal axioms (well, there’s more then two) depend on their informal versions.
In reality, the point of writing down axioms is to capture concrete examples of what your trying to study. Of course, one can do this by writing down the axioms either informally or formally–that is, either in a language lacking a formal semantics or one with a formal semantics. Either way, once you have written down a set of axioms you have in essence captured the objects you wish to study–they are the objects that satisfy the primitive terms of the axioms. So, when someone asks a mathematician, “just what are numbers”, the standard answer is, “they are anything that satisfies the axioms that codify the things I study”. This of course isn’t a definition of numbers–they have no such thing–but it does give meaning to what the mathematician says when she states something like "All natural numbers … ".
Hence, without some sort of axioms your subject matter remains in limbo. For example, an astute high school student may ask their teacher, “Just why is it the case that 2+2=4”, or maybe ask “Just why is it the case that such and such rules for adding fractions hold”. The correct answer, the only answer that can be given, is that those things are so because they are true of the objects we wish to study in the parts of mathematics we call arithmetic. We know that that is true because we have some axioms that specific just what the objects of study in arithmetic look like (The Peano axioms).
As for your point about complex numbers, they–or, rather imaginary quadratic number fields in general–help us prove certain facts about integers, for sure. That does not mean though that those same questions aren’t “resolved” by the axioms. This is a distinction I was hoping not to get into–and avoided in my last long post–but there is a difference between entailment–the idea that all models that satisfy one statement satisfy another statement–and provability–the idea that from some fixed set of axioms and some rules of inference one can generate another statement. While in the last analysis (no pun intended) one may not be able to prove certain facts about integers without recourse to more sophisticated facts about imaginary quadratic number fields, it still is the case that, say, the second-order Peano axioms entail all those facts. Now you may ask what good entailment is, but that is a long discussion in itself. Suffice to say that entail is not only a theoretically important notion but one with much practical important in various situations.
Another good point to bring up is that those quadratic number fields that you bring up have their own set of axioms! Linking back up to my original point, if it weren’t for ring/field axioms and the associated definitions for quadratic number fields, the entire method of complex analysis wouldn’t be available to use in the first place. This theory–these axioms and definitions–are needed to get the subject matter off the ground. So, even if one cannot prove certain facts about integers which generally require complex analysis from the Peano axioms alone, one can certainly prove them from the Peano axioms plus the ring/field axioms plus the definitions of quadratic number fields.
The point is that however you cut it, there’s no escaping axioms in mathematics–even when you think they aren’t at play, they are. They certainly are needed for more than just proving the limitations of our theories. [/quote]
[quote]BlakedaMan wrote:
I think I do actually understand the points of axioms and I wasn’t arguing with you at all, I was simply making the point that the specific set of axioms does in fact show the limits of real integers. Your last paragraph even agrees with that point.
[/quote]
But what do you mean “the specific set of axioms does in fact show the limits of real integers”. What are the limits of “the real integers”? I’m very confused.
If by “the integers” you refer to some class of isomorphic models that satisfy the second-order peano axioms with suitable definitions to handle negative integers, then we are limited in what we can prove from these axioms, and thus about integers, not by some inherent limitation of the axioms, but rather by the inherent incompleteness of second-order logic. If second order logic were complete–that is, if every valid formula of second-order logic was provable–then there would be no limitations on which truths about integers we can derive. For every statement about the integers is either true or false (valid or invalid), and since if second order logic was complete we could prove all the valid formulas, we’d be able to prove all the truths about integers. The fact that we cannot prove all the truths about integers though isn’t some limitation of the second-order Peano axioms–from them every truth about integers is either valid or invalid–but rather a limitation of second-order logic itself–you can’t prove all the valid formulas.
So, to put it bluntly, if P is the conjunction of all the second-order Peano axioms, then “if P, then T” is valid if and only if T is a true statement about “the integers”. So, the second-order Peano axioms tell us every fact about the integers, there are no limitations to be found here.
You could respond “Yeah, but if we stick to deductively complete theories then we can’t find some axioms that only pick out the integers, so this time our knowledge of the integers isn’t limited by the deductive system itself but instead by the axioms.” But this was exactly my point in the other post, that doing this doesn’t show us that somehow our knowledge is limited, it only shows us that our axioms weren’t formed in a descriptive enough language.
If ultimately, as I think, your point is that there simply is no axiom system which picks out the integers and from which alone we can prove all the truths about the integers, then this point doesn’t say anything about the “limitations” of our knowledge unless you require that we can’t “know” something unless we can “prove” it in the formal sense of proof–i.e., a proof is a string of formula each of which is either an axiom or follows from others in the string by some set of uninterpreted rules. This is absurd though–no one would say that our knowledge is limited to what formal proofs can be given for. Virtually no proofs in mathematics are formal proofs. Most are informal model-theoretic proofs about validity. Perhaps your confusion is over the notions of proof and validity and how they relate to axioms and models.
I just don’t see in what sense you can say that axioms tell us anything about the limitations of our knowledge.
EDIT: And about my last paragraph, say “if P, then T” is valid in second-order logic, were P is the Peano axioms and T some truth about integers. It might be that one can’t prove T from P, and that one needs to assume, say, Q–some set of axioms about quadratic number fields and whatever else you need. So then even though one can’t prove T from P, one can prove T from P and Q. This doesn’t mean that somehow our axioms were limited–“if P, then T” was valid, remember. It just tells us a fact about algorithms and formal deductive systems.
[quote]stokedporcupine8 wrote:
duffy,
I apologize for making that so long. Hopefully you find it engaging enough to wade through. [/quote]
I was honor bound to wade through it!
But seriously, it was all quite right and the beat down I deserved for trying to defend an overly broad statement. Nevertheless I’m glad that you can see where I was coming from “in spirit”.
…your “main research area” definitely trumps my bit of background…
[quote]duffyj2 wrote:
stokedporcupine8 wrote:
duffy,
I apologize for making that so long. Hopefully you find it engaging enough to wade through.
I was honor bound to wade through it!
But seriously, it was all quite right and the beat down I deserved for trying to defend an overly broad statement. Nevertheless I’m glad that you can see where I was coming from “in spirit”.
…your “main research area” definitely trumps my bit of background…[/quote]
ah, I hope it didn’t come off like a beat down, that wasn’t my intent, ha. More like a friendly chat?
Anyway the point you were making is an important and interesting one, I wish more people were interested in this or new more about it.
[quote]stokedporcupine8 wrote:
I am, yet again, amazed at some of the stuff people discuss here. Does anyone who posted in this thread actually have enough background in physics in order to understand the research surrounding dark matter and energy, or is everyone just blowing hot air based on some discovery channel documentary?[/quote]
Watch yourself. I innocently suggested something to that effect in the aliens thread and was attacked by a gang of 15 year old cyber bullies. 
LOL.
As I was saying in the other thread, I truly like these discussions and the topic, but there is a real danger in it being pointless when you start discussing theories that you don’t fully understand. “Pop Theoretical Physics” as I call it is so dumbed down that there is a real danger of taking a dumbed down concept and running off in other directions with it. Anyway, I enjoy the discussion. We have some very bright people around here.
This shit is so over my head, although its fun to try and read and tell myself im smart enough to understand it on some minute level :). Agree /w above post of “we have some very bright people around here”
[quote]ah, I hope it didn’t come off like a beat down, that wasn’t my intent, ha. More like a friendly chat?
Anyway the point you were making is an important and interesting one, I wish more people were interested in this or new more about it. [/quote]
Well, I tried to cover myself with an argument that I knew wasn’t entirely perfect. It would have worked with just about anyone else!
Unfortunately for a topic like this you need people who are both interested and possess an adequate background knowledge… and those are few and far between. Even most of the maths students I know have already geared themselves towards the finance sector. It’s sad, since maths is one of the most pure things that mankind possesses.
Where are you coming from incidentaly? Pure maths? I’m a final year theoretical physics student myself.
[quote]duffyj2 wrote:
Unfortunately for a topic like this you need people who are both interested and possess an adequate background knowledge… and those are few and far between.[/quote]
When you mentioned Godel’s incompleteness theorem, I didn’t really think you had a clue what you were talking about.
[quote]duffyj2 wrote:
Even most of the maths students I know have already geared themselves towards the finance sector. It’s sad, since maths is one of the most pure things that mankind possesses.[/quote]
Yes, but you have to remember that most students aren’t going to go into research. Even if they’re reasonably interested in maths, they’re doing the degree to get a good job. But it would be nice if they were all passionate about learning pure mathematics.
It’s up to the department to give the students opportunity to study these things, instead of just doing loads of statistics (which gets more funding in the UK).
[quote]duffyj2 wrote:
Where are you coming from incidentaly? Pure maths? I’m a final year theoretical physics student myself.[/quote]
When you say final year do you mean final year as an undergrad or as a grad?
I was in philosophy/math/physics as an undergrad–now working on my phd in philosophy, my specialty, as I mentioned, is in mathematical logic.
I thought about going into physics, but I realized that given where I was coming from my chances of getting into a good physics program were almost nill. Plus I never focused on it, my focus was always on logic. So, I went into that. I sort of get the best of both worlds now though, since I can always sit in on physics classes and whatnot.
[quote]Rational Gaze wrote:
duffyj2 wrote:
Even most of the maths students I know have already geared themselves towards the finance sector. It’s sad, since maths is one of the most pure things that mankind possesses.
Yes, but you have to remember that most students aren’t going to go into research. Even if they’re reasonably interested in maths, they’re doing the degree to get a good job. But it would be nice if they were all passionate about learning pure mathematics.
It’s up to the department to give the students opportunity to study these things, instead of just doing loads of statistics (which gets more funding in the UK).[/quote]
Unfortunately for us as a society, the greatest practical advances come only after significant theoretical advances–which requires time and money to be put into theoretical research. There are many examples of this, not the least of which is the development of modern electronics. It’s hard to imagine any of it happening without the serious advances of theoretical physics, to say nothing of all the serious advances of mathematics required to get those.
So, sadly, as a culture we value pragmatic advances and put little if any time into theoretical work, not realizing that the former depends on the latter.
Of course people need jobs, so they go were the money is, but that’s the point–as a culture we don’t allocate money effectively, despite what our capitalist friends on this form think. It’s clear what the last two revolutions in physics–Newtons and the revelation of the early twentieth century–gave us, just think what the next revolution will yield.
I was going to write something rather lengthy but I don’t think it would have ended up saying much. As a physics major I had the opportunity to take a course on Cosmology which went over the fundamentals of our theories and understanding of the formation of the Universe. All I can really say is that Astrophyics is about the most math intensive physics disciplines in all of phyics, since to understanding anything you need to be able to meaningful derive solutions to “Field Equations”, basically a bunch of numbers that allow us to model the shape of the Universe. Pretty much no one, physicists included, knows anything about these subject unless they are Phds in Astrophyics. Our professor looked like Gordon Freedman from Half Life so I was inclined to believe what he said.
We studied the solutions to these Field Equations in some detail and they have great predictive power, strongly rooted in observation of astrological events. Some key events to search for if you need some convincing, red-shift and the expansion of the universe, our measure of the Cosmic Background Radition, Lensing effects on light by gravity, the list goes on with theories on gravity that have many experiments that help validify the fact that we have gotten something right.
Anyway all these facts are strongly linked with our “current” model of the Universe which is based off this one equation which models how matter/energy creates the shape and behavior of the Universe. Both Dark Matter and Dark Energy, much like visible matter, radiation and gravity have very specific properties and amounts that are all part of this equation that allow us to accurately model observed behavior.
We have obeserved visible effects of Dark Matter such as light being bent by large unseen objects surrounding visible galaxies. Dark Energy is more mathematical as is currently thought of, to my best recollection, as a kind of energy constant who’s primary property is a repulsive gravitational effect. Basically the Universe is expanding, which is a visible event, and for us to explain that there needs to be x amount of “Dark Energy” which we will define as having properties such as a repulsive gravitational effect to combat the combined force of gravity from all measured visible and dark matter. Dark Energy is strongly rooted in theories about how matter came to be in partical physics and the creation of the Universe. Google Higgs Constant for more of a mind fuck because without a Higgs Partical we cant explain why particals that make up everything has mass, even though we can predict atomic collision out to the ninth deciaml place in equations…
So I would say Dark Matter and Energy are a little better than place holders, they have measurable effects on the Universe and specific properties. There are just gaps that still need to be filled that could change some of the particulars. If they turned out to be complete BS, which would be cool, then expect to see theories like General Relativity and Partical Physcis get totally shit canned are redone from the ground up.
First, to clarify: I do not think it correct to consider a "giveaway " or “giving money to the rich” tax cuts or tax credits which comprise reduction of amount of taxes needed to be paid (as opposed to the concept of paying a check to someone who pays no taxes as supposedly as being a “tax cut”).
And I am generally in favor of people having less money taken from them in taxes. Particularly if their is a socially-useful reason why this should be done in their case.
That being out of the way now (as there are people that have the mindset that reduces taxes comprises GIVING money to the person) it could make sense to have tax credits for publishing scientific work in peer-reviewed journals. Perhaps the tax credit could be related to the readership of the journal in which the work was published. Not directly proportional, but certainly on average, an article published in Science is going to be of more general importance than one published in Dung Beetle Physiology Annual Letters.
As it is, pretty much only academics have financial motivation to do so – publication advances their academic careers.
Once out of academia, publishing isn’t so attractive.
For example, there are several interesting papers that were surely publishable that I could have done after leaving UF and working for Biotest. Example: correlation of solubility paramaters, molecular weight, and melting point with transdermal flux.
It would have been new, interesting, and useful.
But what was in it for me? Nada.
So, having other things to do, such things did not happen.
But you’d better believe if I’d gotten a nice tax credit those suckers would have been done pronto.
[quote]stokedporcupine8 wrote:
duffyj2 wrote:
Where are you coming from incidentaly? Pure maths? I’m a final year theoretical physics student myself.
When you say final year do you mean final year as an undergrad or as a grad?
I was in philosophy/math/physics as an undergrad–now working on my phd in philosophy, my specialty, as I mentioned, is in mathematical logic.
I thought about going into physics, but I realized that given where I was coming from my chances of getting into a good physics program were almost nill. Plus I never focused on it, my focus was always on logic. So, I went into that. I sort of get the best of both worlds now though, since I can always sit in on physics classes and whatnot.
[/quote]
Undergrad. I was asking you about your background since you mentioned the group theory approach to quantum physics. Apparently, this is what my math project of this year consists of. I must admit that I have no idea what I am getting myself into… but I suppose that all the fun is in the finding out.