[HN Gopher] The number line freaks me out (2016)
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The number line freaks me out (2016)
Author : mananaysiempre
Score : 115 points
Date : 2025-02-16 13:50 UTC (3 days ago)
(HTM) web link (mathwithbaddrawings.com)
(TXT) w3m dump (mathwithbaddrawings.com)
| cognomano wrote:
| Related: https://math.stackexchange.com/questions/462790/are-
| there-an...
| kragen wrote:
| Recent and related is the discussion of "Dedekind's subtle
| knife": https://news.ycombinator.com/item?id=43084200
| routerl wrote:
| It 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 wrote:
| >>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 wrote:
| I 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 wrote:
| >>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 wrote:
| > 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.
|
| I think it's more than that... we come with some built-in
| heuristics for probability, which mostly work pretty well.
| Until they don't.
| rbtprograms wrote:
| i would argue our built in heuristics for probability are
| pretty bad, which is why the monty hall problem is so hard
| for most people to grasp (even though it is a relatively
| straight forward application of probability). probabilistic
| thinking comes much less naturally to the human mind than
| deterministic thinking.
| xboxnolifes wrote:
| I find that the bad intuition on the monty hall problem is
| mostly due to the small delta of going from 1 in 3 doors to 1
| in 2 doors, combined with some bad human intuition. If you
| change it to start with 1000 doors, I find it to be a lot
| more intuitively convincing.
| stavros wrote:
| Same here, although I've talked to people who were equally
| confused with that formulation.
| fenomas wrote:
| > when it is actually about relations
|
| I think of it more as, math is ultimately about _symbols_.
| Like, if a mathematician says that "2 = 2" is a true
| statement, a reasonable onlooker might ask "Does that mean that
| all twos are interchangeable? Or that there's a unique concept
| called two and it equals itself?" And the mathematician
| replies, "Neither! It means that the string of symbols '2 = 2'
| is reducible to the symbol 'true', given certain axiomatic
| symbolic transformations. Nothing more, nothing less!".
|
| And obviously we can project concepts onto the symbols, like
| "integer" and "real number", and talk usefully about them, but
| those are the map and the symbols are the terrain, as it were.
| At the edge cases where we're not sure what to think, we have
| to discard the concepts and consult the symbols.
| kragen wrote:
| Platonist mathematicians do exist.
| dr_dshiv wrote:
| The vast majority of mathematicians are platonists. They
| think math is real.
| kragen wrote:
| They might not admit it if they're not drunk enough.
| Meneth wrote:
| And the logical conclusion of that: https://en.wikipedia.
| org/wiki/Mathematical_universe_hypothes...
| fenomas wrote:
| I'll edit out the part of my comment where I suggested they
| don't.
| kragen wrote:
| You can't, so you didn't.
| rini17 wrote:
| And then there are undecidable statements and their ilk.
| atoav wrote:
| I recently thought a student who "always was bad at math" how
| to use it to make his marionette puppet servo controlled. On
| the way we encountered a lot of scaling ranges, working with
| angles, trigonometric functions for motion synthesis, all kinds
| of complex remappings with saturation functions, random walks,
| linear interpolation functions and so on.
|
| He was absolutely stunned and asked me why mathematics wasn't
| thought that way all the time. Instead of a bunch of things he
| had to do, he came to see it as a toolbox with things you can
| use.
|
| And I myself wonder why the hell my maths teachers failed at
| making this easier as well. I distinctly remember my math
| teacher wbo failed to answer me when I asked after months of
| solving integrals why we need those. I had to figure that out
| myself, pre-internet.
| rich_sasha wrote:
| > The vast majority of people think mathematics is about
| numbers, when it is actually about relations
|
| It is somewhat unfortunate that mathematics is two different
| things, simultaneously very closely related and very different.
| One is the abstract study of relationships between axiomatic
| entities, and the other is arithmetic.
|
| Vast majority of people out there need only arithmetic, and boy
| they really need it. Calculating tax, taxi fares, shopping
| bills, splitting bills etc. And to some extent, you need the
| abstract maths to understand arithmetic.
|
| We have one curriculum for that vast majority of people and for
| the few who move on to academic maths. Simplifying ideas like
| integers to number lines doesn't seem like a high price to pay.
| gosub100 wrote:
| > 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
|
| Childrens' brains are not fully developed. I see no gain from
| telling a 6 year old that "most numbers aren't countable".
| Especially because most numbers are never used or interacted
| with in any way shape or form. It's not "lying", it's
| separating concepts and prioritizing.
| krisoft wrote:
| > 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.
|
| I mean maybe? Depends on what your definition of being naive
| about probabilities is. The Monty Hall problem has a sordid
| history of even very learned mathemathicians specialising in
| probability getting it very wrong. For example Paul Erdos got
| it wrong[1] (until someone walked him through it)
|
| Now maybe you count Erdos as someone who is naive about
| probability. In which case I guess you are right. But that puts
| the bar very high then.
|
| 1: https://sites.oxy.edu/lengyel/M372/Vazsonyi2003/vazs30_1.pdf
| sgt101 wrote:
| Like Art is about making pleasing artifacts whereas it's really
| about making business connections and laundering money?
| viraptor wrote:
| > We've only done this because teaching is hard, and thus
| expensive.
|
| That's just silly. We've done that to make the math useful and
| possible to teach. Unless you're saying you're able to start
| with sets of numbers and defining a ring for kids, before
| explaining what 1+1 is.
| behnamoh wrote:
| > 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 wrote:
| Not 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 wrote:
| In what way is dark matter not mysterious? It's a mystery, so
| mysterious seems like the perfect description...
| behnamoh wrote:
| mysterious [?] magic
| Trasmatta wrote:
| I didn't say it was magic
| bawolff wrote:
| > 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.
| pjbeam wrote:
| Maybe intended as wordplay with scrutable, which is a word?
| temp0826 wrote:
| It's a perfectly cromulent word
| genewitch wrote:
| Is a cromulent word a cromule?
| __MatrixMan__ wrote:
| Nothing so spooky deserves to be called a "Real" number.
| Animats wrote:
| The 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 wrote:
| I like to think of continuity as always having more
| "resolution" available if a sharper "picture" is required. It
| is _weird_ though.
| Animats wrote:
| Constructive 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 (2 _N+1) / (2_M).
| dilyevsky wrote:
| Integers are already infinite and you need infinity to express
| things like convergent sequences. Real numbers are a whole new
| level of wtf altogether
| erehweb wrote:
| See https://math.stackexchange.com/questions/4216831/what-
| does-c... for some discussion on convergence using
| constructive mathematics.
| throwaway81523 wrote:
| Well sure, there is a constructive subset of the topic we call
| "mathematics", just like there is a subset that admits only
| numbers less than or equal to 5. But if you want the whole
| topic of mathematics, it's going to include nonconstructive
| theorems like the Banach-Tarski paradox. One could take a
| philosophical view for or against the idea that the non-
| measurable set in the paradox platonically "exists", but either
| way, those theorems are a legitimate part of mathematics. At
| best you can say the theorems are about mythological entities
| rather than "real" ones.
| jasdi wrote:
| Searching for meaning is all good but sometimes its just the
| umbrella man - https://www.youtube.com/watch?v=yznRGS9f-jI
| BobbyTables2 wrote:
| Even "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 wrote:
| I 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 wrote:
| Was 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 wrote:
| We 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 wrote:
| I 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 wrote:
| It 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 wrote:
| I 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 wrote:
| I 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_
| numbe...
| gcanyon wrote:
| Wait, the number of programs is countable? Are we saying that
| programs must be of finite length? (because if not a diagonal
| approach would prove them to be uncountable)
| titanomachy wrote:
| "Countable" as used in mathematics does not necessarily
| imply finite. The integers are "countably infinite", and so
| is anything you can put in a 1:1 correspondence with
| integers.
| gcanyon wrote:
| But as I said, if programs are of infinite length, then a
| diagonalization argument proves the computable numbers
| not to be countable I think.
| titanomachy wrote:
| Ah yes, of course you're right. I think it does make
| sense to assume the programs are finite, I don't think
| numbers described by an infinite program should be
| considered computable.
| Gabriel54 wrote:
| Yup! Programs are assumed to have finite length, in the
| sense that the program must have a finite description. Of
| course, it may use recursion or include a loop that runs
| forever, for example.
| dllthomas wrote:
| You can't point to one of them, but you can point to infinite
| sets of them.
| shric wrote:
| Not all non computable numbers are undescribable. Chaitin's
| constant[1] is non-computable but can be described.
|
| [1] https://en.m.wikipedia.org/wiki/Chaitin%27s_constant
| red75prime wrote:
| > In what way do they exist?
|
| In a way that makes the real number line continuous. Those
| numbers have to be there if we want the set to have properties
| useful for practical applications like algebra.
| ngruhn wrote:
| As far as I understand, if you look at a number between 0 and 1
| with a truly random infinite decimal expansion:
| 0.22134967842153005356...
|
| then there is absolutely no pattern in the digits, so a program
| that wants to compute it can do no better than storing all the
| digits. But then the program would have infinite size.
| jstanley wrote:
| I can write a program that will visit every number between 0
| and 1, but it will take infinitely long to run and use infinite
| memory.
| RyanAdamas wrote:
| Math 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 wrote:
| I'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 wrote:
| You 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 wrote:
| I did mention objects, not numbers. Maybe "symbols"
| representing concepts. I'd go with "name" for both zero and
| one -- nonexistence and existence.
| dataflow wrote:
| > 1 is the only number. Everything else is a name.
|
| http://scihi.org/leopold-kronecker/
| awesome_dude wrote:
| I'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 wrote:
| Well, you're wrong. An irrational number can't be described by
| a fraction. That's the very definition.
| cafeinux wrote:
| They 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 p as p /1 and give you a rational representation of p.
| throwaway81523 wrote:
| "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_irr...
| dr_dshiv wrote:
| You 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 wrote:
| Not 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 wrote:
| You 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 wrote:
| Actually, 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 wrote:
| > 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 wrote:
| This helps explain the different kinds of numbers in that recent
| HN article about how hard it is to make an accurate calculator.
| Spivak wrote:
| One 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 wrote:
| There 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 wrote:
| You mean "NOT even definable"?
| nialv7 wrote:
| Oops. Thanks for pointing out.
| vishnugupta wrote:
| > 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 wrote:
| An 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.)
| Dylan16807 wrote:
| The set of text files is clearly countable because it's made
| of binary. Do you think you can make an English sentence that
| can't be written into a text file?
| rvense wrote:
| If humanity lives forever, it will keep on inventing new
| words and therefore new sentences. So the question of
| whether or not language is finite is really the same
| question as whether or not the universe is.
| krisoft wrote:
| Finite is not the same thing as countable.
| https://mathinsight.org/definition/countably_infinite
| Dylan16807 wrote:
| > If humanity lives forever, it will keep on inventing
| new words and therefore new sentences.
|
| That just increases the fraction of text files that count
| as "English". Which doesn't affect the argument.
|
| > the question of whether or not language is finite
|
| does not need to be answered. If English has a thousand
| words and never gains another one, the list of English
| sentences is countably infinite. If English gains 10%
| more words every year forever, the list of English
| sentences is still countably infinite.
| sgt101 wrote:
| eventually that will be something that is not engish in
| any way we, but formally it will not be english straight
| away
| truckerbill wrote:
| You are right because you can recursively add clauses.
| 'Buffalo buffalo...' or 'This was my dad's dad's dad's...' If
| you think you have a full set you can always add one more
| sambapa wrote:
| You seem to misunderstand the concept of countable infinity
| truckerbill wrote:
| Haha oh you're right, was misframing in my head the word
| countable as meaning finite.
|
| I even provide the definition of countable infinity in my
| counterargument without realising it, though maybe that
| too is a misunderstanding.
| water-data-dude wrote:
| See The Library of Babel, by Jorge Luis Borges:
|
| https://archive.org/details/TheLibraryOfBabel
| tacitusarc wrote:
| Why would this follow? There are many more ants and yet I can
| use a single word to refer to them, and certainly the biology
| text that defines them contains fewer words than there are
| ants.
| xigoi wrote:
| You can certainly refer to the set of all real numbers, but
| that's not the same as referring to one particular number.
| gcanyon wrote:
| This video parallels the article, going further. I love the tag
| line, "We know none of the numbers."
| https://www.youtube.com/watch?v=5TkIe60y2GI
| mannyv wrote:
| There are an infinite number of numbers between each number on
| the number line.
|
| That's always something fun to think about.
| rob74 wrote:
| ... and the article doesn't even mention the concept of +/-
| infinity and the fun things you can do with that!
| sambapa wrote:
| And those two numbers are boundaries of an interval bijective
| to the whole number line...
| left-struck wrote:
| I think it's slightly more fun to say there are an infinite
| number of numbers between each number on the INFINITE number
| line
| bandrami wrote:
| Another 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 wrote:
| Are 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 wrote:
| Nope. 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 wrote:
| f(x) = |x + i|
|
| Non-symmetric real-valued function on C.
| bandrami wrote:
| And 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 wrote:
| Seems 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.
| xigoi wrote:
| 1 and -1 are distinguishable: one of them equals its
| square, the other does not.
| cgriswald wrote:
| Sure, but I'm not sure I'm understanding the argument. I
| don't understand how a function like f(z) = e^z has to be
| symmetric about the real number line or how i and -i
| _aren 't_ distinguishable with something like Im(z) > 0.
| Is there a proof somewhere I can read?
| bandrami wrote:
| It falls out of complex numbers satisfying the conditions
| of a field though I don't know of a specific "proof" of
| that (you generally don't "prove" definitions). You could
| equally say "i is indistinguishable from 1/i" or "i's
| additive inverse is its multiplicative inverse"; in
| either case it's an arbitrary choice which of the
| conjugates is positive and which is negative. The key
| being that you cannot say "i > -i" because of that.
| xigoi wrote:
| Expanding the definition of the imaginary part, this says
|
| (x - x*) / 2i = 1
|
| where x* denotes the complement. If you replace i with -i,
| the graph will be precisely the complement of the original
| graph.
| praptak wrote:
| From 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 wrote:
| When 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 wrote:
| Probabilistically 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 wrote:
| What 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 wrote:
| Do 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.
| n4r9 wrote:
| This is not the case. A set A can be said to be larger than set
| B if there exists an injection from A to B, but not from B to
| A. This is a well-defined extension of the concept of size in
| finite numbers, and preserves all the properties you might
| expect (e.g. transitivity).
| watwut wrote:
| Concept of comparison exists for infinite numbers. That is not
| controversial, bit more of 101 math analysis on university.
| zkmon wrote:
| If you say one kind of infinite is smaller than the other
| kind, then the first kind no longer qualifies to be called as
| an infinite as it smaller than some other number. So first
| you need to define what an infinite is.
|
| Also infinite is not a number. And comparison exists only for
| numbers.
| n4r9 wrote:
| The finite numbers extend easily to "cardinal numbers",
| which may be infinite:
| https://en.m.wikipedia.org/wiki/Cardinal_number
| jstanley wrote:
| This is not right.
|
| There are infinitely many integers.
|
| There are infinitely many real numbers between each pair of
| integers.
|
| Thus there are more real numbers than integers.
| InDubioProRubio wrote:
| Its 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 wrote:
| I wonder if this was the inspiration for this numberphile video?
| https://www.youtube.com/watch?v=5TkIe60y2GI
|
| The video does go further than the article.
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