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The number line freaks me out (2016)
- cognomano 2y agoRelated: https://math.stackexchange.com/questions/462790/are-there-any-examples-of-non-computable-real-numbers https://math.stackexchange.com/questions/462790/are-there-an...
- kragen 2y agoRecent and related is the discussion of "Dedekind's subtle knife": https://news.ycombinator.com/item?id=43084200 https://news.ycombinator.com/item?id=43084200
- jheriko 2y ago[flagged]
- routerl 2y agoIt seems to be an article about all those "harmless" lies we tell students. The vast majority of people think mathematics is about numbers, when it is actually about relations, and numbers are just some of the entities whose relations mathematics studies. Nobody is born with this misconception; we teach it, and test it, and thereby ingrain it in the minds of every student, most of whom will never study mathematics at a level that makes them go "wait, what?". The overwhelming majority of people never get to this level. I suspect this is also why statistics feels so counterintuitive to so many people, including me. The Monty Hall problem is only a problem to those who are naive about probability, which is most people, because most of us don't learn any of this stuff early enough to form long lasting, correct instincts. It's not fair to students to bake "harmless" lies into their early education, as a way to simplify the topic such that it becomes more easily teachable. We've only done this because teaching is hard, and thus expensive. Education is expensive, at every step. It's not fair or productive to build a gate around proper education that makes it available only to those who can afford it at the level where the early misconceptions get corrected. Even those people end up spending a lot of cognitive capital on all those "wait, what?" moments, when their cognitive capital would be better spent elsewhere.
- kamaal 2y ago>>when it is actually about relations, and numbers are just some of the entities whose relations mathematics studies. Well no. Math is studying anything at their most atomic machinations. That mostly involves. - Making hypothesis, that is assumptions about start conditions and rules of play. - Evolving the system you just created. Such that conclusions are consistent with the rules of play. The real deal is good math involves lots of paper work, to an extent you could almost say Math is a writing skill than a thinking skill. Think of it like generating a lengthy changelog.
- SyzygyRhythm 2y agoI would say instead that math is a game. A universal game with no predefined rules at all and only one guideline: if the rules you make up lead to a contradiction, then the rules are probably boring. If your rules say that 1+1=3, then you can prove anything and the whole thing becomes uninteresting. Mathematicians have come up with various rules (axioms) that seem to work pretty well. And they spend a great deal of time figuring out their consequences. But it may still happen that the rules have a contradiction and they need to come up with a different set. Sometimes mathematicians add extra rules when they run into a roadblock. And part of the meta-game is to come up with the minimum set of extra rules they need to keep going. Sometimes they spend time figuring out if the existing rules aren't needed.
- kamaal 2y ago>>I would say instead that math is a game. Yup, and as you keep going the level too go up! But the core ideas are simple though- 1. Start some where where you understand things enough to make sense. 2. Make the smallest possible, atomic change to some aspect of thing you know at point 1. 3. Test if the change sticks- If yes, repeat steps 1 - 3 4. If the change doesn't stick- Go back to step 1. Now either make a different change to the same thing or make a new change to a different thing. Repeat steps 1 - 3. As you can see you write a lot. Like really a lot. Math is just writing skills.
- mcphage 2y ago
- behnamoh 2y ago> Simon Gregg calls noncomputable numbers “the dark matter of the number world”... Thanks but no, we don't need more of this kind of bs naming. "dark matter" already ruined physics because it implies something mysterious and magical is going on whereas it's quite the contrary. I hate it when people dumb down beautiful abstract concepts to the point that it's not only not intuitive, it actually makes the thing less accessible to those who are not in the know.
- danielmarkbruce 2y agoNot just abstract concepts. We dumb down moderately complex concrete things to stupidly simple explanations and names too. It's soooo bad. We basically encourage each other to have hazy 30,000 ft view of things when the details aren't really that difficult, it just takes a little work.
- Trasmatta 2y agoIn what way is dark matter not mysterious? It's a mystery, so mysterious seems like the perfect description...
- bawolff 2y ago> It should be a timidating Total nitpick, but i think the in in intimidating means "into a state of being timid" and not "in" in the sense of opposite of timidating.
- __MatrixMan__ 2y agoNothing so spooky deserves to be called a "Real" number.
- Animats 2y agoThe underlying problem is that infinity doesn't exist. It's a convenient illusion to make special cases go away. It's possible to have entirely constructive mathematics. In a true constructive model, everything can be constructed in a finite number of steps. There are only integers, no reals.
- pjbeam 2y agoI like to think of continuity as always having more "resolution" available if a sharper "picture" is required. It is _weird_ though.
- Animats 2y agoConstructive mathematics can handle rational numbers. Rational numbers are continuous, in the sense of being infinitely sub dividable. That is, between N/M and (N+1)/M lies (2N+1) / (2M).
- dilyevsky 2y agoIntegers are already infinite and you need infinity to express things like convergent sequences. Real numbers are a whole new level of wtf altogether
- erehweb 2y agoSee https://math.stackexchange.com/questions/4216831/what-does-constructive-mathematics-say-about-limits https://math.stackexchange.com/questions/4216831/what-does-c... for some discussion on convergence using constructive mathematics.
- kreetx 2y agoIt's the definition of "exists" that is contested. To me, the infinity of anything is a generator (a program that continues to print digits). Does some infinite number exists "now"? Not really (it's being printed). But is the number being printed larger than any given integer? Yes. So, does infinity exists? ¯\_(ツ)_/¯ "The Emperor's New Mind" is a great book on this and related topics.
- jasdi 2y agoSearching for meaning is all good but sometimes its just the umbrella man - https://www.youtube.com/watch?v=yznRGS9f-jI https://www.youtube.com/watch?v=yznRGS9f-jI
- BobbyTables2 2y agoEven “e” and “pi” would have been noncomputable at one point in time. But the noncomputable numbers make me wonder if our notion of mathematics is too general/powerful.
- Aerroon 2y agoI think in this case it's Reals that are too general. What is the virtue of these uncomputable numbers? If we can't compute/express them then what can we do with them?
- Dylan16807 2y agoWas pi ever really uncomputable? You can draw a really big circle and measure it in multiple ways. And when e was defined as a symbol, it was with a computation, (1 + 1/n)^n
- throwaway81523 2y agoWe call a real number computable if there is an algorithm that can compute it to arbitrarily high precision. So e and pi have always been computable.
- ziofill 2y agoI find it hard to wrap my head around non computable numbers. How can I even “point to one” of them if I can’t express/describe it? And if I cannot communicate which number I’m referring to, does it really exist? In what way do they exist?
- Gabriel54 2y agoIt is a good question. I can say that, briefly, you have to take two things for granted. 1, that the real numbers can be constructed, and 2, that the number of computable numbers is countable because the number of programs describing them is countable. Therefore there must exist uncomputable numbers (and in fact, 'almost all' real numbers are uncomputable).
- ziofill 2y agoI can accept the second thing, but how do you mean the first one in a way that doesn’t fall back onto the second one? What is the way to “construct” reals that isn’t a program?
- Gabriel54 2y agoI mean construction in the mathematical sense. If you believe, for example, that the rational numbers exist, then it is easy to construct the real numbers. Very roughly, we look at the set of all sequences of rational numbers that converge in a specific way (technically, all Cauchy sequences) and call this the "set of real numbers" (technically, after taking an appropriate equivalence class, since intuitively multiple sequences can converge to the same real number). [1] has a few other constructions. This is very different from a program, which has its own definition. [1] https://en.wikipedia.org/wiki/Construction_of_the_real_numbers https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...
- ziofill 2y agoSo, if I understand correctly, you are saying that many (actually most) Cauchy sequences converge to uncomputable numbers. I think I understand a bit more, though I still have an issue with defining such a Cauchy sequence, because either I can describe it exactly (in which case the number it converges to can have an infinite expansion but it's computable, like pi) or I cannot define it exactly, in which case I don't know what number it converges to. I think my gap is that I don't understand how these constructions differ from programs. By program I mean a sequence of instructions, which may have an infinite number of steps because of loops, but it still has a finite description. I agree that all such programs are countable. Do you mean programs with an infinite description?
- RyanAdamas 2y agoMath is the study of futility. Futility to calculate, to understand, to define, to rationalize. 1 is the only number. Everything else is a name.
- sssilver 2y agoI’d paraphrase. I’d say that zero and one are the only objects. Everything else is a name. So in a sense math is exploration of the relation between existence and nonexistence.
- RyanAdamas 2y agoYou can't show me zero of something which is why you can't divide by it. Zero is a placeholder for what we can't show which is also why negative number exist on the opposite side of it. Zero isn't a number, it too is a name.
- sssilver 2y agoI did mention objects, not numbers. Maybe “symbols” representing concepts. I’d go with “name” for both zero and one — nonexistence and existence.
- autoexec 2y ago> You can't show me zero of something which is why you can't divide by it. I've shown you zero fish. The number of fish I've shown you is zero. If I tracked you down, brought a fish with me and showed it to you, I'd have shown you 1 fish, but I haven't.
- dataflow 2y ago> 1 is the only number. Everything else is a name. http://scihi.org/leopold-kronecker/ http://scihi.org/leopold-kronecker/
- awesome_dude 2y agoI'm probably not very clever - but I don't get why calling the gap "fractions" is problematic. The example the author gives of "fractions" is... rational numbers, and then proceeds to say "what about irrational numbers" - but in mymind (and this is probably where I'm a wrong?) an irrational number is still a fraction of a whole number, just we cannot express it "properly" (yet)
- bongodongobob 2y agoWell, you're wrong. An irrational number can't be described by a fraction. That's the very definition.
- cafeinux 2y agoThey mean "fraction" as in "part", the same way that an arm is a fraction of a whole body. But it's more about the words we use in everyday language than about mathematical definitions. Also, I think I remember that the definition of a rational number implies fractions of integers. Otherwise I could write π as π/1 and give you a rational representation of π.
- throwaway81523 2y ago"x is rational" means there are two integers p,q such that x=p/q. So for example, 2/3 is rational (p=2, q=3), but the square root of 2 is not rational (there is no such fraction). The last part is not very obvious (it greatly distressed the Pythagoreans when they figured it out) but there are a bunch of proofs in Wikipedia: https://en.wikipedia.org/wiki/Square_root_of_2#Proofs_of_irrationality https://en.wikipedia.org/wiki/Square_root_of_2#Proofs_of_irr...
- sadeshmukh 2y agoA fraction is some integer over another integer. Those are not fractions, though they may be in a colloquial sense.
- dr_dshiv 2y agoYou can represent pi on a number line but it is absolutely completely impossible to randomly put a dot down on a number line and have it be pi. You can achieve endless measurable precision with decimal rational numbers. So randomly placing a dot on a line will always be a rational number. Right???
- dhosek 2y agoNot necessarily. If you assume an ideal ruler and compass, it‘s pretty easy to construct the square root of any number which means that you can easily put a dot on that number line which is provably not a rational number. There is the question of whether continuity actually exists (are time and space quantized like matter and energy or are they continuous? this is currently unknown) and the fact that your paper and the line on it are composed of discrete molecules. But if the real numbers are, in fact, real, then the probability that your dot is at a rational point is actually 0 since while the number of rationals in [0,1] is infinite, it’s only countably infinite and the number of irrationals in [0,1] is uncountably infinite meaning |ℚ|/|ℝ\ℚ|=0.
- raincole 2y agoYou mean in real world or in math? In real world, correct, assuming "dot" and "number line" are consist of real world materials. In math, you need to define "randomly placing a dot" first, because it's proven there isn't a uniform distribution over real numbers ("pick randomly" is usually a colloquial way to say "pick from a uniform distribution.")
- GrantMoyer 2y agoActually, the opposite is true. The rationals have measure zero, so a number selected uniformly at random from 0 to 1 (formalized way of placing a dot on a number line), has a 0% chance of being rational.
- housecarpenter 2y ago> it is absolutely completely impossible to randomly put a dot down on a number line and have it be pi It's not impossible, it just has zero probability of occurring.
- apt-apt-apt-apt 2y agoThis helps explain the different kinds of numbers in that recent HN article about how hard it is to make an accurate calculator.
- Spivak 2y agoOne thing not mentioned that still gets to me is that all the numbers we know, that we will ever know, is measure zero on the real number line. The lovecraftian nightmare numbers, the infinite maw of the unknowable, aren't some rare exception, they're everything.
- nialv7 2y agoThere are more structures in non-computable numbers. As an example, a non-computable number could be "definable", i.e. you can describe it. For example, a real number the nth digit in the binary representation of which equals to whether the nth Turing machine halts. Since the set of all English sentences is countable, whereas there are uncountably many real numbers, it follows that there must be numbers that are NOT even definable. Think about that.
- rwoerz 2y agoYou mean "NOT even definable"?
- nialv7 2y agoOops. Thanks for pointing out.
- vishnugupta 2y ago> Since the set of all English sentences is countable Is it? Where can I read a proof? I have a feeling it’s uncountable set but would be happy to see a proof one way or another.
- housecarpenter 2y agoAn spoken English sentence is a finite string of phonemes. The set of allowable phonemes is finite. Given a finite set X, the set of all finite strings of elements of X is countable. (The latter statement holds because for any given n, the set X_n of all strings of length n is finite. So you can count the members of X_0, then count the members of X_1, and so on, and by continuing on in that way you'll eventually count out all members of X. You never run out of numbers to assign to the next set because at each point the set of numbers you've already assigned is finite (it's smaller in size than X_0, ..., X_n combined, for some n). In fact, even if you allow countably infinitely many phonemes to be used, the X_n sets will still be countable, if not finite, and in that case their union is still countable: to see that, you can take enumerations of each set put them together as columns an a matrix. Even though the matrix is infinite in both dimensions, it has finite diagonals, so you can enumerate its cells by going a diagonal at a time, like this (the numbers reflect the cells' order in the numeration): 1 3 6 10 15 2 5 9 14 4 8 13 7 12 11 However if you allow sentences to be countably infinitely long, then even when you only have finitely many phonemes, the set of all sentences will be uncountable, because in that case each countably infinitely long sentence can be mapped to a real number represented as an expansion in some base, and you can apply Cantor's diagonal argument. The "just count out each X_n separately" argument doesn't work in this case because it only applies to the sentences of finite length.)
- gcanyon 2y agoThis video parallels the article, going further. I love the tag line, "We know none of the numbers." https://www.youtube.com/watch?v=5TkIe60y2GI https://www.youtube.com/watch?v=5TkIe60y2GI
- mannyv 2y agoThere are an infinite number of numbers between each number on the number line. That's always something fun to think about.
- rob74 2y ago... and the article doesn't even mention the concept of +/- infinity and the fun things you can do with that!
- sambapa 2y agoAnd those two numbers are boundaries of an interval bijective to the whole number line...
- left-struck 2y agoI think it’s slightly more fun to say there are an infinite number of numbers between each number on the INFINITE number line
- bandrami 2y agoAnother number line mind-blown moment is that the complex plane is actually a half-plane since the distinction between i and -i is arbitrary, so any graph in the complex plane has to be symmetric about the real number line.
- red75prime 2y agoAre you sure? The distinction between i and -i is no more arbitrary than the distinction between 1 and -1. Example of an asymmetric graph: Im(x) = 1
- bandrami 2y agoNope. Take any true mathematical sentence and (consistently) replace i with -i and it remains true; that is not the case for 1 and -1. Im(x) = 1 is meaningless; it would have to be Im(x) = ±1. (In fact you'll only ever see complex numbers in the form of a±bi, never a+bi alone.) It's why you can't say e.g. -i < i; the signs on purely imaginary numbers are not an ordering.
- red75prime 2y agof(x) = |x + i| Non-symmetric real-valued function on C.
- bandrami 2y agoAnd indistinguishable from f(x) = |x - i| The choice of one as +i and the other as -i is arbitrary, which is not true with 1 and -1.
- cgriswald 2y agoSeems pretty true with 1 and -1. Map R with f(x) = -x and f(x) = |x - 1| for x in your new mapping is indistinguishable from f(x) = |x + 1| in R. In any case I’d say this is arbitrary like using + for addition and - for subtraction. It seems like you’re just talking about the symbols themselves. I’m not sure how you get to half plane from there.
- praptak 2y agoFrom what I remember "things you can compute" splits into "things you can mathematically define" and "things for which you can construct a Turing machine which computes them". If we define "x is computable" as "there exists a Turing machine T(x) which takes n as input and produces n-th digit of x" then there are numbers which are defineable but not computable.
- zkmon 2y agoWhen you say "most" of the numbers are non-computable, the word "most" is meaningless in this context. There are infinitely many of each kind of numbers you have listed there. You can't compare one infinity to another and say that one kind of infinite is bigger than the other. The concept of comparison (smaller/bigger) doesn't exist outside of finite numbers. Cantor was just what people thought he was - a crack, who did not consider the bounds of logical comparison. And for the line itself, the line is not made up of numbers. Line is made up of continuity, while numbers are cuts in that continuum. Infinite number of cuts do not make up a continuous piece. Mathematical continuity (or extent or measure or span) is the essence of the imaginary spatial existence. It is not composed of cuts. A cut is a non-existence, completely opposite of the existence.
- stouset 2y agoProbabilistically it’s true. Pick a number at random from the uniform distribution of values between zero and one. With nearly total certainty, this value cannot be represented except by enumerating every single one of its infinite digits.
- alkonaut 2y agoWhat does it mean to pick a number at random between zero and one? Does "picking a number at random" even make sense for an infinitely large set where you can't describe most items? Isn't "picking" the act of describing an item? (This might sound like a stupid question but I'm sure there is a mathematical definition of "picking") If I pick a number at random using some method for picking that requires me to identify what I picked then 100% of the time I'll get a number I can identify, such as the number that is the solution to x^2=2, or the ratio between a square and a circle, or the quotient of 3 and 7. All those numbers I can't describe will never be picked. I can do infinitely many coin flips and say the number I picked has the decimals described by that binary sequence. But I'd never be done picking...
- rossant 2y agoDo you know measure theory? It gives you a formal definition of "almost all" or "almost surely" based on subsets which have the same measure as the full set they're in. Like the irrational numbers between 0 and 1.
- InDubioProRubio 2y agoIts because its recursive in nature- every segment contains a new line and every segment of that line contains again - a infinite set of sub-segments. Infinity in a box, right in front of your numeric microscope. Which is why dividing by zero- is exactly the same operation. You take something finite- and you unpack the boxes- in parallel. Every time the operator hits something finite, it unpacks a new set of parallel boxes. The sum of all the boxes, is a infity with a signature. And those parallel running overlapping infinityssquences, form the irrational numbers
- globular-toast 2y agoI wonder if this was the inspiration for this numberphile video? https://www.youtube.com/watch?v=5TkIe60y2GI https://www.youtube.com/watch?v=5TkIe60y2GI The video does go further than the article.
- Justta 2y agoJust forget number line after high school. Number line is needed for engineering graphs and economic data
- bbstats 2y agoYes, there are different kinds of numbers.