How Classical Cryptography Will Survive Quantum Computers

Discussion in 'privacy technology' started by ronjor, Dec 30, 2017.

  1. RockLobster

    RockLobster Registered Member

    Joined:
    Nov 8, 2007
    Posts:
    1,215
    Yes π does have a precise definition as the ratio of the circumference of a circle to its diameter, the problem is, π cannot be expressed precisely in mathematical terms which was my original point. Math cannot describe a circle using π because no matter how many decimal places you use, it is never quite accurate.
    You could say
    π = 3.14159 but if you examine the circle πd microscopically you would find the ends don't quite line up.
    You could then define π to a thousand decimal places but microscopically the circle circumference still would not line up exactly. It would be close, but not exact.
    If you use the equation of a circle
    x^2+y^2=r^2 to define the circumference, no matter how many millions of points you make there is, microscopically gaps between the points. If you join them with lines you create a polygon. It may be microscopic lines but it is still polygon with millions of sides because no math exists to create the curve between those points except by using π and as we know already, π is an irrational that cannot be precisely defined.
    Therefore math can only approximate a circle.
     
    Last edited: Jan 13, 2018
  2. Stefan Froberg

    Stefan Froberg Registered Member

    Joined:
    Jul 30, 2014
    Posts:
    284
    And not only computers but pretty much anything in the real world.
    Laws of math might be precise but physical world never is. It's just not possible.
     
  3. reasonablePrivacy

    reasonablePrivacy Registered Member

    Joined:
    Oct 7, 2017
    Posts:
    208
    Location:
    Some country in the European Union
    "ratio of the circumference of a circle to its diameter" - this statement based on precise mathematical terms. π is a precise mathematical term.
    I think you have too narrow imagination what is a mathematical term. The fact that something is not precisely expressable in actual numbers doesn't mean it is not a mathematical concept, term.

    All you are stating are problems when someone wants to apply this mathematical term in other part of human knowledge for example computer engineering.
     
  4. RockLobster

    RockLobster Registered Member

    Joined:
    Nov 8, 2007
    Posts:
    1,215
    Yes I understand exactly what you mean by that.
    Ok let's put it another way
    If you do not give π a numerical value and you say π is the ratio of the diameter of a circle to its circumference, that is fine to say, conceptually π represents that undefined ratio.
    But, as soon as you give π a numerical value, then, clearly π is not the ratio of the diameter of a circle to its circumference. It is almost, but not.
    Math says that does not matter but I think it does when you want to use math to describe the universe.
    That is to use math in an abstract sense where numerical values that don't quite fit, are considered irrelevent and there is a school of thought amongst some mathematicians that says we are tailoring math to make it fit things it doesn't.
     
    Last edited: Jan 13, 2018
  5. Yuki2718

    Yuki2718 Registered Member

    Joined:
    Aug 15, 2014
    Posts:
    1,262
    You can google around w/ "photon", "interference", "double slit" or such. Unlike black body radiation, it won't require advanced math to understand (tho explanation is another thing), but in short single photons will be plotted like particle but after thousands of trials those plots make interference fringes. From my understanding of your post #22 I think it can't be explained, but you can try. Other physics fan might wonder why I didn't mention Compton scattering as it shows light have momentum which might be more characteristic to particle, but it doesn't change the conclusion that light is not particle in classical meaning. Well, possibly I had to say that quantum physicist use the term particle in different meaning, i.e. sth countable rather than sth like a ball.
     
    Last edited: Jan 13, 2018
  6. Yuki2718

    Yuki2718 Registered Member

    Joined:
    Aug 15, 2014
    Posts:
    1,262
    @RockLobster @reasonablePrivacy
    The fact is both of you are right. Sure, math is cleverly constructed not to be bothered by error. In math there're many ways to define π and you don't need to define it as the ratio of circumference to diameter, but whatever def you use, there's no problem of error. But it doesn't mean math do not include ANY sense of error or approximation. But the meaning or error/approx in math is completely diff from usual sense. Remember how you learned integral. You covered a graph w/ rectangles which have error for section, then you diminished the error by using narrower and narrower rectangles, and defined the section as its limit. Note "lim_{n → ∞}f_n = f" doesn't mean f_n becomes f when you increased n. It just mean you can diminish error whatever degree you want, however, the error will never be 0 (we use ε-δ for precise argument). Put it in another way, in math there're 2 use of "=", the one is equivalence and the other is just a definition, e.g. "0.99999... = 1" comes from the latter. When limit is closed (belongs to the same set which f_n belongs to), mathmatician can construct sth new upon it, but they never confuse that notion. One way to define π is "2∫_0^1 dy/√(1 - y^2)", you see it uses limit in 2 ways, the one is improper integral and the other is integral itself. Regardless of Riemann integral or Lebesgue, it includes error in certain sense.

    As to the question whether math can describe circle, a hard problem exists in another aspect. You know a circle can be defined as a set of points in R^2 whose Euclidean metric from certain point are all equal, also axiom of metric and def of Euclidean metric, right? OK, then how those infinite (yeah, uncountable infinite) points can make continuous curve of circumference? You may even know Cauchy sequence and completion, but were you really satisfied when you leaned it? I was not. That seemed to be technique which cleverly bypass the hard problem by, yes, ε-δ. My boss when I was undergraduate said to me, no body really knows what the "continuous" means. I think that's true, not to mention CH, Banach-Tarski, and some other strange "pardoxes" (I'm not saying paradox in mathematical meaning). It's no matter as long as you just want to use math for sth useful, but if current composition of Euclidean figure is good enough is another matter, just like Peano axioms have serious problem to describe natural numbers.

    I'm afraid we're going more off-topic. :(
     
    Last edited: Jan 13, 2018
  7. RockLobster

    RockLobster Registered Member

    Joined:
    Nov 8, 2007
    Posts:
    1,215
    @Yuki2718 excellent explanation, you made some points I had not considered I will look them up and yes we did get a little off topic but still, it is all relative in a roundabout way.
     
Loading...