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Mathematical purity vs real world understanding

  1. Jul 5, 2011 #1
    This is a very philosophical question, so i understand the answer is going to be very arbitrary and subject to opinion.

    I am interested in both Pure mathematics and Theoretical Physics. The time is soon approaching, where i must decide upon which path to take.
    I love proofs, and i love rigor, yet sometimes i find myself sitting in a number theory class and wondering, why? What is the point? Sure its beautiful and i enjoy it, but there is still a nagging feeling of doubt, like i am missing something. When i am learning physics I experience the reverse, i wonder "Is this too applicable, Am i just being a glorified engineer?"

    My goal in entering science is to understand the universe. So of course Physics makes sense, but some mathematicians claim that if you understand mathematics, you understand all possible universes, and thus it sounds much more important and interesting then theoretical physics. My worry is, that if i take a PHD in mathematics, i would not understand say Quantum Field theory or Advanced Relativity as well as a Physics Grad student. This leaves me at a loss! I wish i could somehow combine the mathematical proof with the physical understanding. Alot of people mention mathematical physicists, but from my research they seem to be primarily interested in mathematical technique over understanding, say how a certain partial differential equation works, rather then how quarks interacted at the beginning of time (just an example). My dilemma is that either path i choose i will lose something, either rigor, or real world understanding.

    It all depends on the nature of theoretical physics. Is physics just mathematics, that we choose to apply to an experiment or is there something more? Am i confusing the map with the landscape?
  2. jcsd
  3. Jul 5, 2011 #2
    If i were to take a mathematical Phd i would like to work in Topology, Differential Geometry or Algebraic Geometry. If i went down the Physics path i would work in High Energy Particle Physics.

    ^Just providing either course of action
  4. Jul 6, 2011 #3
    no one want to help? : (
  5. Jul 6, 2011 #4
    Usually I wouldn't answer in threads like these because I'm not exactly an expert in this but seeing that you haven't gotten a reply I'll give it a try.. I think you haven't got a fast reply because its moreso of your personal opinion. You have to delve deep into yourself and see what you really want.

    Its hard to tell you what YOU like more. Personally it sounds like you are more interested in the physics of things *understanding the why* but torn apart by the phrase of mathematicians understanding all possible universes. <- I don't know much about this very statement persay.

    I don't know what level math and physics you are up to but.. Advice I would give to you (and advice I have gotten) is to take a mixture of both and see what your interest leans as you self-explore in the interaction with the subject.

    And you keep saying that physics does not have rigor. Where did you get that from?
  6. Jul 6, 2011 #5
    Thanks for answering!
    Yeah i do see this as a personal dilema, i was just hoping for some guidance (hence the forum:biggrin:). When you say Physics explains why, that is what i used to believe. I used to see mathematics as a tool for physics, gradually though i began to see that physics is just a subset of mathematics, and if we keep asking "why?" something happens, we will go deeper and deeper into physics until we pop out into pure mathematics. Hence, if you want true understanding, head for pure mathematics?
    As for Physics lacking rigor, i have heard that from so many mathematicians i simply accepted it as fact (which may not be good). There is alot of hand waving in physics, this method gives us the right answer, so it must be true, mathematicians would be more intersted in why it is true, would they not?
  7. Jul 6, 2011 #6
    My only issue is that alot of mathematicians take no interest in physics at all. Some proudly proclaim how little physics they have taken. Yet it was Quantum mechanics and Relativity that got me interested in science in the first place, so if most mathematician's lack an understanding of those subjects it puts me back on square one. I am aware there are some mathematical physicists who are aware of both, but do they trade something off for their mathematical understanding? I have the nagging suspision that alot of mathematical physicists take more interest in applying differential equations as objects of their own interest rather than understanding, say, space time. But this of course leads back to my reductionism, in that number theory is closer to the essence of truth then even relativity would be... And i keep going in circles!
  8. Jul 6, 2011 #7
    I studied pure mathematics in undergraduate, and did a Masters under the Department of Mathematics but my thesis topic is on topological black holes in anti-de Sitter space. Now I am pursuing a PhD in astrophysics, though my research is mainly theoretical physics, that sounds confusing doesn't it :biggrin: Anyway, ultimately there is no one stopping you to choose whatever way you wish to "understand the universe", you can still take courses in physics department and read physics textbooks while being a math major and vice versa, so I don't see why you have to feel uneasy about choosing either math or physics - you do choose one, but it doesn't mean you completely disregard the other. When there are interesting courses in the math department, such as mean curvature flow or global differential geometry being offered, I will still take up the courses even if I am no longer in the math department.

    Personally, I like to have mathematical rigor whenever possible. For example, in general relativity, you can learn via the physics route through books like https://www.amazon.com/Gravity-Intr...sr_1_1?s=books&ie=UTF8&qid=1309934262&sr=1-1", which one you prefer of course largely depends on your personality. But sometimes you don't have a choice, for example, there is no complete rigorous foundation for quantum field theory available. So it's good to be able to think like a physicist even if you end up choosing pure mathematics. The two fields might be closely related, but I don't think it is fair to call physics a "subset of mathematics", they require very different thinking. In time, you will see. :tongue:
    Last edited by a moderator: Apr 26, 2017
  9. Jul 6, 2011 #8
    I think you need to first of all nail down exactly what you mean by 'understanding the universe.' To me, 'understanding the universe' mostly boils down to two questions:
    1. Why am I here?
    2. How does stuff work?
    To answer the first question, I turn to studying things like philosophy and religion. To answer the second, which I think is more relevant to your question, I turn to physics. I think that that is, essentially, the goal of physics: to understand how the universe works on a physical level (perhaps quantifying it could be added to that).

    Pure mathematics, I see as being in a different light. I think it's a little bit of a stretch to say that math allows you to understand all possible universes. I see pure math as being more similar to music and art than to physics; I don't see it as a way to understand the universe. Look at music: Why do we play music? Why do I spend hours upon hours practicing the ability to get strings to vibrate at certain frequencies in certain sequences? It's not because it has an inherent 'usefulness,' but simply because it's beautiful. Music is beautiful, and beauty can help us to see the universe in a completely different light. However, I would argue that the goal is not to understand the universe. The goal is, quite simply, beauty. I think the same goes for pure mathematics. The goal is not to understand the universe, the goal is to understand or create something beautiful.

    Now personally, my chosen field of study lies somewhere withing Applied Math, Physics, and Electrical Engineering. I like to try to understand why the things around me are the way they are. Pure math just really doesn't quite do it for me. When I study science, I want to try to understand the world around me, and I find that pure math (math that's not applied to anything) doesn't help me do that. The problem is, you're not me. Hopefully this gives you some ideas to think about, but at the end of the day, it's your choice that really matters. Try to figure out what you *really* mean when you say "I want to understand the universe"
  10. Jul 6, 2011 #9
    Yes that sounds very interesting! But its what i would miss out on that gnaws at me the most. I am currently double majoring in mathematics and physics, but if i went down the physics path in grad school i would not be able to take courses in graduate number theory, set theory or any of the more abstract graduate courses, why? Because i will be too busy learning physics courses. I am not saying that i prefer either at this stage, but i feel as though mathematics and physics is just so big no man can understand all of it anymore. If i went down the physics path i may end up 5 years latter thinking back on how i could have learnt some really beautiful mathematics, and if i did the same with mathematics, i would feel as though my understanding of the workings of our universe was lacking. There just does not seem to be enough time, it makes me wish i lived in gauss's day.
    And what is thinking like a physicist really? Is it just waving your hand at rigor, saying "we don't really need to know that"? or does it involve more geometric perspective?
    Last edited by a moderator: Apr 26, 2017
  11. Jul 6, 2011 #10
    I used to think like that, that physics explained how stuff worked and mathematics was just a logic game, but having read much of the thoughts of prominent mathematicians and physicists, i was caused to rethink my position. What can be our ultimate understanding of the universe? say it is a couple of equations and constants that you could write down on a page, why those equations? we will never know... Unless you accept the multiverse perspective and our universe is a part of the anthropic subset, which of course leads us to an infinite number of universes, all governed by different laws of physics, but all describable by mathematics, thus why wouldn't pure mathematics be the ultimate theory, the real "how" answer? I know that multiverse theory may not be right, and probably is not even testable, but if it is, then mathematics is the real underlying branch of knowledge and thus worth devoting a lifes research to.
  12. Jul 6, 2011 #11
    You're making a very big assumption in saying that mathematics is the underlying explanation of everything. How do you know that math isn't just a close approximation of reality that happens to work out beautifully within itself? If that's the case, then the idea that mathematics is the underlying explanation for the universe falls apart completely.

    But suppose your assumption is correct. I still really think you need to define what you mean by 'understanding the universe' more clearly. You can go on forever asking the "why?" question. For example:
    Why does an apple fall?
    Gravity pulls it down.
    Why is there gravity?
    Masses attract each other.
    Why do they do that?
    [suppose some fundamental equation explains this]
    Why does that equation work?
    [explain with pure mathematics, as you've suggested]
    What do you base pure mathematics on?
    [perhaps you say something about axioms]
    Why should I accept those axioms?

    Do you see how it can just go on forever? You have to pick a point that satisfies you. Personally, I would be perfectly satisfied with a list of equations and explanations that describe the physical workings of the universe. Maybe you wouldn't be, and that's fine, but you should understand that you need to stop somewhere. Even pure math is based on relatively arbitrary axioms, why should those be the right axioms?

    Unfortunately, you just have to figure out what *part* of understanding the universe is the most satisfying to you. Trying to understand the universe in its entirety is a never-ending quest that will probably just frustrate you more than anything. It's almost a meaningless quest. What does it even mean to understand entirely? There's always another "why?" that you can ask.

    Don't forget, too, that physics and pure mathematics are HUGE subjects as it is. You're eventually going to have to decide which tiny portion you want to understand deeply because you're not going to be able to understand it all. It sucks, yes, but it's reality unfortunately. By all means pursue an understanding of everything, but you should be willing to be happy with the little portions of understanding you do achieve.
  13. Jul 6, 2011 #12
    I too was reluctant to respond to this thread considering my experience lies almost entirely within the field of pure mathematics.. that withstanding:

    You're right about this, you're going to have to make a choice of specialty. If, as you say, what you really want is to understand the universe, than I think you should study theoretical physics. Moreover, if you think that studying mathematics will somehow allow you to understand all possible universes, with our universe as just one particular manifestation among many, you're going to be disappointed.

    I think that what a mathematician would do with our universe is examine its geometry, extract its intuition and distill out a set of axioms and definitions that the mathematician feels appropriately capture the idea of an arbitrary geometry. From here the mathematician would make conjectures about all intended consequences of these axioms, which he or she would turn into theorems through the application of proof. So I suppose in this manner the mathematician creates his or her own universe, but it would be the idyllic platonic one, cold and austere.

    True, but I think that from the mathematician's POV the differential equation is more interesting than the quark, and feels just as real, if not more so, considering that quarks are subatomic particles which are only indirectly observable.

    From what I understand there are some areas of physics that are so abstract that they are generally worked on by mathematicians instead of physicists, but it seems like if these areas are to be considered science than they would still have to be amenable to experimental verification, so I'm not sure how much 'proof' factors in to even the most abstract areas of theoretical physics.

    I'm not sure number theory really admits a lot of physical intuition, so maybe geometry/topology would strike a better balance.

    Personally, one reason I've always valued the pursuit of pure mathematics is that while physics can be appreciated at the level of an informed layman, mathematics cannot.
  14. Jul 6, 2011 #13
    I did not claim that there was an axiom of section of pure mathematics in which everything else is based (except maybe fundemental areas such as set theory). What i contended was that all possible universes, being infinite, would be explained by pure mathematics. I see this as a half ground between the post modern view you have taken, that seems to be, we can never understand everything, everything is subjective, so just choose a small little niche. I personally cannot abide by that. If i wanted that, i would be taking english literature or some liberal arts junk.
    While mathematics won't provide a definitive answer, it will provide the medium to understand. we may never achieve it, but it is the journey not the reaching of the destination which i think matters.
  15. Jul 6, 2011 #14
    “Physicists investigate one cosmos, but mathematicians can explore all possible worlds." - Shing Tung Yau "The shape of Inner space".
  16. Jul 6, 2011 #15
    I think S.T.Yau quoted someone else for that... By the way have you read and consider the advice by mathematical physicist John Baez available http://math.ucr.edu/home/baez/advice.html" [Broken] somewhere down the page under the section "math or physics"?

    In theoretical physics research, as you can read on hep-th or gr-qc on arXiv, most of them are far from rigorous from a mathematician's standard, for example there is no rigorous proof for the validity of AdS/CFT correspondence, yet it has been applied to various problem like condensed matter system. If everything needs the rigor of a mathematician, progress will be very slow, physics does not work like that, after all there is the need for verification from experiment or observations. In the words of Frank Wilczek I believe, "In physics, your solution should convince a reasonable person. In math, you have to convince a person who's trying to make trouble. Ultimately, in physics, you're hoping to convince Nature. And I've found Nature to be pretty reasonable".

    Coming to your worry about not being able to learn as much as you would like, well, that happens to everyone. Eventually you choose to learn everything about something, and something about everything.
    Last edited by a moderator: May 5, 2017
  17. Jul 6, 2011 #16
    Yes Yang did quote someone else, Clifford Taubes a mathematical physicist at harvard. Anyway semantics aside, i am inclined to agree with him. If you want to understand gravity you have to understand differential geometry, if you want to understand electromagnetism you have got to understand vector calculus and if you want to begin to understand quantum mechanics you have to learn about hilbert spaces etc. So if you want to understand physics, you must understand math... I understand my point is a reductionist one, but i think it is valid, if you want to understand a branch of physics you must understand math, and to understand the reasoning behind that math, you must understand more abstract mathematics, I see the connection going from Astrophysics to Relativity to Geometry to Algebra and number theory and finally to set theory and fundemental axiom construction. Physics reminds me more and more of how i used to view engineering. It is a great career for progressing technology and of the highest importance within society, but withing engineering classes, they tend to plug and chug formula and accept things without wondering why. I have a deep desire to research the most fundemental knowledge that humans may access and the more i think of it, the more it leads to pure mathematics.
    Last edited by a moderator: May 5, 2017
  18. Jul 6, 2011 #17
    By no means to i want to claim that engineering or physics is useless, far from it, sometimes i worry that they are too useful to the real world for me to make a career out of them.
  19. Jul 6, 2011 #18
    No, while it is true that if you want to understand those fields you have to learn the maths it isn't true that you have to get a full blown rigorous treatment of the maths to understand everything about the physics. Why? Because when you come up with formulas it isn't about testing every combination of letters to try to come up with something which fits, instead you have a conceptual picture in your head on how you think that it should work and then you try to write that down mathematically. That picture is the physics, the formula is just a representation. It doesn't matter how well you understand the formula if you don't get the picture, you still don't understand the physics.
  20. Jul 6, 2011 #19
    It doesn't matter which path you'll take. You can decide to do physics, but still take many classes in mathematics and read many math books. Or you can do mathematics, and take classes in physics and read textbooks.

    In my opinion, it's better to actually do physics. If you do physics then you'll train your physical and mathematical intuition. And you'll see why some math thingies are important. Only doing mathematics gives you no intuition about physics at all.

    On the statement that mathematicians study all possible universes. I'm a PhD student in mathematics and I have no clue what the physical interpretation is about what I'm doing. So I might study all possible universes, but I have no clue what these universes are and how to understand them. If you want to understand the universe, then do physics.
  21. Jul 6, 2011 #20
    But why do you say that? The goal of pure mathematics is not to understand the universe. The goal of pure mathematics is to study math for the sake of math. Maybe it's true that pure math will explain everything, but I see no reason to believe that, and I think you're going to be disappointed. What if it's true that there is no possible way to perfectly describe the universe using mathematics? I think that's a perfectly fair question. In that case, I think mathematicians would continue to study math, and math-based physics would become like Newtonian physics is today: a useful approximation, but not correct.

    I don't think everything is subjective in the sense that two people could start with the same set of basic assumptions and come to completely different or contradictory, but equally correct, conclusions. However, I think the basic set of assumptions is indeed arbitrary. I just think that you have to pick a place to start. That's all I really meant. Based on your other replies to this thread, I think I misinterpreted what you meant. I thought you were seeking an understanding of the universe that made no assumptions at all.

    If you want to be a part of the journey to understand the universe, then you should study physics. The ultimate goal of physics is to explain the universe. The ultimate goal of pure mathematics is *not* to understand the universe, but to understand mathematics. Yes, physicists use mathematics a lot, but the mindsets are completely different. If you want to be part of the journey to understand the universe, then I think physics is the way to go, because physics is trying to understand the universe. Pure mathematics really isn't. I think studying pure math based on the assumption that it is what will eventually provide the perfect explanation for the universe is just going to disappoint you. It could very well be that pure mathematics is completely unable to describe the universe. It's still your choice, and you should study pure math if you want to, but if you want to be part of the journey to understanding the universe, then I think you should study the subject that's trying to understand the universe: physics.
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