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Being Hyperanalytical

  1. Apr 16, 2012 #1
    I have a problem and I feel that it is becoming something of a psychological disorder. Whenever I am learning some scientific subject, I have a desire to "prove" all the ideas "from first principles". At some times it has escalated to an unavoidable desire to prove something so obvious that even a child would understand it at first glance. I feel this is reaching a level of almost "abnormality".

    I have trouble making progress in my subjects unless I can prove the ideas whose proof I did not yet encounter. For the more difficult topics I usually take the idea for granted and write it down somewhere hoping to see the proof at some other time during my life, but for the simpler ones that are just out of my reach, I try to prove them but end up becoming frustrated over my efforts and often it is very time consuming. Many a time I am not successful in my efforts, and in the process people probably think I'm mad doing what it is I try to do.

    I will soon study engineering and I feel that as an engineer, I will have neither the necessity nor the opportunity of being so skeptical about what I learn. For example, it's hard for me to use a theorem from math in physics unless I know its proof etc. but then that just makes my work all the more difficult.

    Does anyone else have my problem? Is my problem a problem or is it a good thing? How can I manage it? I assume all mathematicians/physicists/economists/computer scientists faced my problem at some point in their careers, but how do they deal with it?

    BiP
     
  2. jcsd
  3. Apr 16, 2012 #2

    micromass

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    You sound exactly like a mathematician!! :tongue2:

    I'm a bit amazed that you want to study engineering. If you go for engineering, then you'll have to set aside this characteristic of yours. Engineers don't bother with proofs, as long as the math works out. If you can't set aside this part of you, then you might not make it through engineering.

    Have you thought about a math major? Mathematics is the ideal major for somebody who wants to prove everything from first principles.

    Don't feel abnormal though. I know how you feel. When I was in high school, I was trying to prove simple things like "addition of integers is commutative". I never quite was able to prove it (because I essentially didn't know what integers were exactly).

    Is it a good thing or a bad thing?? As a mathematician it's a good thing. As an engineer it's a bad thing. It all depends on the circumstances.
     
  4. Apr 16, 2012 #3
    Thanks for your fast reply micro!
    What about for an economist/computer scientist?

    I was originally planning a math major, but I am more toward the applied math. I am not a big fan of stuff like analysis and topology, which I could do as a hobby, but would just not prefer taking classes in them.

    What bothers is me is when I cannot prove ideas mathematically as they are seen in other subjects, such as physics,chemistry,economics and computer science.

    I am really in a pickle deciding what I want to do!

    BiP
     
  5. Apr 16, 2012 #4
    +1 for what micromass said

    Perhaps physics would be more your kind of thing than maths or engineering, it's got the 'real life' factor that engineering has along side first principles and proofs.
     
  6. Apr 16, 2012 #5

    micromass

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    Why would you prefer not taking classes in them? Do you think that it's useless? (just a question)

    I have no experience with economy. But computer science could also be something for you. You could also start from first principles there and move on. Sometimes, however, the math isn't very "kosher" (in my experience).

    Physics might also be something for you. Although I always found it hard to justify why a certain piece of math could be used in that physics problem.
    As I see it, most sciences use math because it's useful. Thus they only focus on the useful aspects of it. Things like "proofs" and stuff, are not useful, so they're a waste of time. If you want to see proofs of things, then you're almost forced to take courses in the math department.
     
  7. Apr 16, 2012 #6
    I was under the impression that graduate level work in any scientific field is proof-based, whether it be physics, economics, comp sci etc. not just math. I might have been wrong.

    I don't find any piece of knowledge to be useless, it's that I find analysis and topology to be much more difficult and usually I cannot manage difficult things under the time and space pressure of a classroom setting. As a hobby I would be able to freely pursue it without any stress. Of course in such a case my progress would be slower, but at least my health would not be destroyed by the stress.

    I like physics, but like you, found the math applications in it to be a bit unnerving. For example, I will often wonder why math is useful in physics, even though I see no real explanation for it, and that wonder itself will slow me down in physics.

    Thanks for your help!

    BiP
     
  8. Apr 16, 2012 #7

    micromass

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    Are you still in high school?? Of course it'll be difficult for you. These courses are upper-level math courses and requires some prerequisites. If had to do topology when I was in high school, I would have failed miserably.

    To be honest, if you got the right prereqs and knowledge, then topology and analysis are a walk in the park!! They're not as difficult as everybody says they are.
     
  9. Apr 16, 2012 #8
    What do you mean? Do you know there is a branch of theoretical physics called mathematical physics? Google it! That's what I'm planning to specialize in (I'm currently an undergrad combining math and physics)
     
  10. Apr 16, 2012 #9
    That sounds interesting!!! The only thing I know from that subject is the Buckingham Pi Theorem but I don't really understand it :P

    Micro, thanks for your fast reply again. I'm a high school senior but I still feel as though I would find pure mathematics to be more stressful than applied math.

    BiP
     
  11. Apr 16, 2012 #10
    Become a mathematician.

    I don't understand why you don't find Analysis and Topology interesting? You seem to like to prove things, so analysis might be fun. And if you like to prove things then I find it hard to believe that you wouldn't like either subject. They are pretty interesting subjects in their own.

    And like micromass said, it becomes a walk in the park if you have a good foundation. I don't know too much about Topology, but I saw a part of it that looked a bit like the principles of variation used in Calculus (I might have mistaken this with something else). And Real Analysis just builds up from calculus from a more rigorous standpoint.
     
    Last edited: Apr 16, 2012
  12. Apr 16, 2012 #11
    I have that problem. Actually, studying math provides no refuge from having to use black-box theorems, I have found. It simply would not have been possible to do all the work that I needed to do in grad school if I went through every proof.

    It's not that I don't believe the theorems. I can use a theorem, as I drive a car if I want. I mean, it's not that I want to just be convinced logically. I want to be convinced intuitively. I want all the results to be obvious to me, not just logically proven. A mere logical proof will be forgotten in 2 seconds, so there's not that much value in it, other than to check your work. No, I want to just be able to see at a glance that it's obviously true to the point where a formal proof is almost superfluous. And I want EVERYTHING to be that way. I am one of the most intellectually greedy people I have ever heard of.

    It's more than that. I don't just want to know why everything is true. I also want to know why you would come up with it in the first place. To feel as if I could have invented it myself.

    In some ways, it may be a handicap to be this intellectually demanding, although in other ways it's an advantage. But it's just my nature. Doing it any other way makes me sick. Because things are not set up to be very friendly for someone with this mindset, I end up feeling sick a lot.

    It's not just for mathematicians. Any intellectual should be fairly skeptical and think for themselves, whether it's engineering or physics or math. It's just a matter of degree.
     
  13. Apr 16, 2012 #12
    homeo I think you nailed it on the spot! That's the way I feel too, but when I see proofs in analysis, I feel like there's no way in hell I could even hope to have an intuitive sense of the proof. In applied math and engineering the problem isn't as bad, that's why I feel like fleeing to those fields. I feel that this skeptical aspect of this nature is both a gift and a curse in a most bizzare sense. It's a problem we have that allows us to remember concepts better than others, and yet we take more time than others to absorb the concept.

    BiP
     
  14. Apr 16, 2012 #13

    chiro

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    Hey Bipolarity.

    One thing I want you to realize is that if you have some set of isolated assumptions in a scientific field and you want to prove things about science, then if your assumptions are wrong, no matter how severe, then everything falls over.

    If you're a mathematician, then that is different because the assumptions that mathematicians don't necessarily have to correspond to reality: there might exist assumptions and constraints for particular systems that model reality rather well, but for mathematician many don't really care that it even has to.

    You might be interested to know that many young programmers (including myself at one point in the past) wanted to create all the code from scratch, design everything from scratch and basically do everything so that they could say they could call it without any doubt, entirely work of their own making.

    Problem is that when you get to real world stuff, not only does no-one care, but doing this is just not feasible: in other words you do what you have to do to do whatever it is you need to do and no more and since there's lots of stuff to do, you just have to do a good enough job without having to do everything from scratch (it's got to be good enough so that you don't create bugs and screw up the system though).

    Even in mathematics, I highly doubt that one person proves every single thing they come across because like the above example, it's not necessary. If you understand enough about a particular result so that you can use it and know when not to use it, that should be enough.

    The thing is that all development is incremental and we have already created the wheel many times over and the thing is, at some point in your endeavor, you are going to have to take some things for granted because if you don't do this, you won't get anything done.

    You will at some point have to take a leap of faith and go with your instincts on some things and because of that, it will mean that you will have to choose what to work on and what to ignore: what to trust and what not to trust.
     
  15. Apr 16, 2012 #14
    I understand. The problem then is essentially the fact that I don't live forever and because of that, I have to focus my time on the issues where I can actually work and find enjoyment while taking for granted the ideas which I don't have the time to analyze carefully.

    Now I have entered a state of cynicism, towards nature, for not letting me live forever. I would deeply have enjoyed analyzing everything to the fullest extent, but nature has created an upper bound on what I can learn and I don't know what to say. I feel angry now towards the fact that I'm not immortal and that in end I have to end up choosing some knowledge over others... it is tough for me to extinguish the more broader aspects of my curiosity simply because of this stupid biological constraint on my life.

    But I thank you all for supporting me in this realization. This realization, I might say, is the single biggest shock that I have ever experienced. Particularly because it intrudes upon the very reason for my existence... but I guess it was kind of inevitable. Well, good fight, life.

    Would it not be fun if our lives were long enough to "know" everything?

    BiP
     
  16. Apr 16, 2012 #15
    Actually, analysis was part of what made me flee from engineering. I could get an intuitive sense of the proofs, I found. It just takes some practice and preparation. In principle, engineering should actually be a better fit for people like us because math is so complicated and huge that you have to know massive amounts of it to be able to create new math. That leads to a strong incentive to use black-boxes in order to save time. If I went and worked for the power company now, which I might actually go do when I'm done with the PhD, all the math would be trivial and obvious, so I wouldn't have any problem. I would be able to understand everything completely. Very simple compared to what I am doing now. But, in practice, engineers often don't think that way, so classes for engineering students can be very disappointing. That's what caused me to quit electrical engineering and study math.

     
  17. Apr 16, 2012 #16
    I can relate.

    However, there really IS a genuine advantage to wanting to understand things for yourself. The problem is, if you don't reinvent the wheel, then you will be stuck with the same wheel forever. THAT is dangerous. We have to have people who are willing to question what's been done before and change it.

    I think of it this way. You should be as flexible as possible. So, insisting that you have to understand everything is compromising your flexibility. It limits your options. So, it should be on the table as an option, should it happen to be the most effective thing to do. But it shouldn't be over-used, either.
     
  18. Apr 16, 2012 #17

    chiro

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    Well in terms of you only having one life, this is something that has been and still is being debated ever since someone else saw someone physically die. If I were you, I wouldn't let this cloud your judgement, or even demoralize you from making any progress or investigation no matter how significant or otherwise you may think it is. Your question will most likely be answered when your cross that bridge and 'go west' as they say.

    With regards to knowing everything, I am actually glad that there many (possibly infinite) things to learn. The world would be boring if we knew everything.

    One example I would like to use relates to the story of the Devil. I'm not a religious person myself (although I do believe in the existing of a creator who has done a wonderful job! Kudos!) but this story transcends any religion and can be seen in every culture, and every walk of life.

    The story goes that someone wishes for something and the Devil appears. The devil says "I will give you everything you want as long as I have your soul". The person usually gets what they want, some having more wiser suggestions than the others, but eventually the list becomes exhausted and they get everything and the Devil gets their 'soul'.

    Now imagine that you got everything you ever wanted and there was nothing else to strive for, discover, find out, or risk. This is contrary to everything that we observe around us: thing's are always changing, new things are always being discovered and quite frankly I think it's great that this is the way things are and I hope and anticipate that although we will learn more and more as time goes by (if we still exist by then), there will always be something new to learn, risks to take, and in the words of Feynman: "The pleasure of finding things out".

    So I would embrace this rather than denounce it, because if human beings had everything they ever wanted all at once in some finite range of what we call 'time', the world would boring and no-one would be happy and in fact I imagine most of us would probably kill ourselves after the fact.

    This kind of thing drives everyone: it drives the mother and the father who work their guts out just in the hope that their children will become something that will surpass them just as the scientist is driven by the uncertainty that awaits them: uncertainty really is the elixir of life and without it we would have no (or not enough) variation and things would grind to a halt.
     
  19. Apr 16, 2012 #18
    Chiro, you've greatly expanded my view on the matter! Thanks! I will learn to be more optimistic about the situation. But to be honest, I do not think I would regret knowing everything in science. If I did know everything, I would be happy that I know it and switch to something endlessly passionate like music or sports that is hopefully invulnerable to math. For example, I do not think that even if you knew all math, you could get bored of eating or playing frisbee. But I feel like these things are not as fun unless you already know all the math :S

    The skeptical aspects of my mind are what originally allowed me to ace calculus and calculus-based physics, but left me with a keen desire to understand analysis (explanation of calculus) and perhaps mathematical physics (explanation of physics). But they themselves would expose me to a multiplicity of problems that would only cause my skeptical side to go crazy.

    I strongly enjoy applied math. How proof-oriented is applied math?

    I have to decide between a top-ranked engineering school and a medium-ranked LAC. The engineering school offers only engineering, but top programs in all its engineering fields and has a good reputation in industry. But the professors there have told me themselves as engineering just might not be my kind of thing if I am curious to see the mathematical proof behind things...

    The LAC graduates very few math/CS majors each year, but those who do so usually make it to medium-ranked grad schools. The LAC is academically less rigorous, and I will be taking a lot of required humanities courses which I have no interest in.

    My dream is to be an applied mathematician, with a strong focus on my proof-background to solving problems in various fields, perhaps cryptography and analysis of the market/banks. I just haven't decided which undergraduate path will take me there nor is there many people to advise me.

    Do people ever go from engineering to applied math? I know the reverse is true, but what about the forward direction?

    As homeomorphic said, I want to have options for graduate school. Perhaps I will really like engineering... it would be impossible at the LAC.

    BiP
     
  20. Apr 16, 2012 #19

    chiro

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    For what you've stated being an applied mathematician, I would recommend you take a healthy mix of statistics, applied mathematics and pure mathematics. If you go on to do postgraduate coursework you can take analysis and topology which should help your appetite for the kind of understanding you seek.

    I would also get comfortable with being able to program a computer in some way and also being able to get up and speak, write reports and even write recommendations to people that are non-technical as this is what many applied mathematicians actually do: in other words, they usually give advice or recommendations based on their technical work to people that need to make decisions based on this advice. Remember that the people that make these decisions are often way too busy and their time is precious which means they need a guy like you to crunch the numbers and give a verdict.

    Also if you want to understand the limits of systems in general I would also do some work on logic and computation as well. Knowing models of computation and limits will give insights into a myriad of things including the most general systems like the ones we find in nature.

    Considering you want to do engineering as an option, I would ask other people to give their advice on this matter but from what you've said it sounds like the focus of engineering might be misaligned to what you have said. This is only a speculative opinion but again based on what you've said about trying to really understand something like calculus (analysis), engineers don't really worry about this kind of thing: they are more worried about a completely different perspective at a completely different level that is more macroscopic in comparison to the kind of microscopic nature that the mathematician would worry about: it's apples vs oranges.
     
  21. Apr 17, 2012 #20
    By the way, math isn't all about "getting things done". It's also for entertainment. If you don't get any pleasure from it, you shouldn't do it. We're humans, not slaves. So, from that point of view, I can justify my own indulgences in trying to understand things, rather than use them to "get things done". The pleasure I derive from math is entirely in terms of being able to understand things, rather than in "getting things done". Any time I put getting things done before understanding, I am making a sacrifice to get the piece of paper at the end.

    If being a mathematician means I have to worry about getting things done and not worry about understanding, then I'm just going to quit because it's just not the right profession for me in that case. I came to understand. I didn't come to churn out as many papers as I possibly could.
     
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