Arthur T Knackerbracket has found the following story:
I remember first learning as a student that some infinities are bigger than others. For some sets of numbers, it was easy to see how. The set of integers is infinite, and the set of real numbers is infinite, and it seemed immediately clear that there are fewer integers than reals. Demonstrations and proofs of the fact were cool, but I already knew what they showed me.
Other relationships between infinities were not so easy to grok. Consider: There are an infinite numbers of points on a sheet of paper. There are an infinite numbers of points on a wall. These infinities are equal to one another. But how? Mathematician Yuri Manin demonstrates how:
I explained this to my grandson, that there are as many points in a sheet of paper as there are on the wall of the room. "Take the sheet of paper, and hold it so that it blocks your view of the wall completely. The paper hides the wall from your sight. Now if a beam of light comes out of every point on the wall and lands in your eye, it must pass through the sheet of paper. Each point on the wall corresponds to a point on the sheet of paper, so there must be the same number of each."
I remember reading that explanation in school and feeling both amazed and enlightened. What sorcery is this? So simple, so beautiful. Informal proofs of this sort made me want to learn more mathematics.
Manin told the story quoted above in an interview a decade or so ago with Mikhail Gelfand, We Do Not Choose Mathematics as Our Profession, It Chooses Us. It was a good read throughout and reminded me again how I came to enjoy math.
(Score: -1, Offtopic) by Anonymous Coward on Friday November 29 2019, @11:38PM
Now get off my lawn and take your bong with you.
(Score: 4, Insightful) by opinionated_science on Friday November 29 2019, @11:54PM (36 children)
informal proofs , are in fact, the whole point of proof.
The mathematical basis for many proofs is to provide an explicit and direct explanation of proofs.
For example, Andrew Wiles's "Fermat's last Theorem" proof, now shows that X^n +y^n=Z^n where n3, for integer solutions.
Hence, in this universe it is impossible to decompose a big cube into 2 smaller cubes, made of cubes of the same size.
Pretty cool huh?
(Score: 3, Insightful) by legont on Saturday November 30 2019, @12:54AM (32 children)
The whole point of math is the truth in any and all universes. That's what other scientists usually fail to grasp.
"Wealth is the relentless enemy of understanding" - John Kenneth Galbraith.
(Score: 0) by Anonymous Coward on Saturday November 30 2019, @01:57AM (1 child)
What about the opposite universe? Anything that is true here is false there.
(Score: 2) by Bot on Friday December 06 2019, @07:01PM
You cannot know, because in that universe "anything that is true in the opposite universe is false here" is false.
Account abandoned.
(Score: 3, Informative) by Immerman on Saturday November 30 2019, @03:34AM (19 children)
Ah, but it's not. Math is a purely mental construct - it only reflects reality insofar as the starting axioms accurately reflect reality, and it's been mathematically proven that it's impossible to prove that they do.
Math is a huge edifice built upon "if that is true then this must also be true". If in another universe something was fundamentally different, e.g. 1+1 somehow equals 3, then the vast majority mathematical "truths" that work in our universe would be rendered completely irrelevant.
(Score: 3, Funny) by KritonK on Saturday November 30 2019, @03:55PM
I remember our math professor proving that 1+1=2, using the Peano axioms [wikipedia.org]. Roughly, the proof went something like this: 1 is a natural number (first axiom). Since 1 is a natural number, then 1+1 is also a natural number (second axiom), so let's call that number, "2"!
(Score: 2) by legont on Saturday November 30 2019, @04:31PM (17 children)
What you are saying is that if say all thintellegence in the universe is dead, the math theorems would be wrong.
Similary, you propose that there might exists a universe where math theorems are different.
Both are wrong.
The second one is simple - math does not depend on the shape of a universe in question. (an opposite might be true, but it's another story)
On the first it is a matter of faith. I believe that math exists independently of conciousness, such as 1+1=2 even after we kill all the humans or even observers. Where are the math facts while all the observers are dead is an open question.
"Wealth is the relentless enemy of understanding" - John Kenneth Galbraith.
(Score: 2, Interesting) by Anonymous Coward on Saturday November 30 2019, @04:57PM (1 child)
Perhaps in another universe if you have one widget and you get another widget and put them together a third widget appears out of nowhere. Hence 1 + 1 = 3 in that universe. Then if you remove one of the three widgets another widget magically disappears in that universe. Hence 3 - 1 = 1 in that universe.
Math does make fundamental assumptions and those assumptions are tied to our observations of this universe. There is no reason to assume that the universe should even follow consistent laws to begin with, we could live in a universe where gravity changes randomly and things appear and disappear out of nowhere for no reason whatsoever with no patterns.
Out of all possible universes a coherent universe is very unlikely. The fact that we can predict a future instance (I throw the ball in the air it will come back down) based on past experience is a very unlikely universe and if the universe was a product of random chance all future instances would not be related to past instances. That is it's possible for us to have patterns at one given time (ie: I throw a ball in the air it comes back down, I throw it up a second time it comes back down. If I throw it up enough times I'm bond to get two or even three consecutive instances where it comes back down by chance but not an infinite amount of them) but if the universe were a product of chance future instances should not be based on past instances and I should not be able to reliably predict that throwing the ball up will result in it coming back down each and every time.
The people that claim 'multiverse' try to say 'well, if you roll the dice enough times you are likely to get this specific pattern'. They miss the point. In a randomly generated string sufficiently large sure I can pick out a sentence if the string was large enough but future characters should not be predictable based on past characters. IE, if I do find a specifically sought out string I should not then be able to predict what future characters come next based on the past characters that I have already observed if the string was random. In this universe I can, I know that if I throw a ball in the air it will reliably come back down each and every time. If the universe is random that should not happen. Multiple instances in a row where the ball does come back down = likely if the ball was thrown up in the air enough times. For the pattern to predictably and reliably repeat itself each time .... impossible for a universe to have formed this way by chance as there are far more possible universes with different future patterns to have formed even if one that had predictable patterns during a given time interval did form.
(Score: 0) by Anonymous Coward on Sunday December 01 2019, @02:05AM
> Perhaps in another universe if you have one widget and you get another widget and put them together a third widget appears out of nowhere.
> Hence 1 + 1 = 3 in that universe. Then if you remove one of the three widgets another widget magically disappears in that universe.
> Hence 3 - 1 = 1 in that universe.
Ah is this the magic synergy they talk about?
(Score: 2) by Immerman on Saturday November 30 2019, @05:14PM (12 children)
The theorems are only as true as the axioms they're built upon. And the axioms are only as true as the accuracy with which they represent the universe. The broadly accepted axioms are all chosen to reflect common sense and observed properties of this universe, but there's no guarantees that they do so accurately. And even if they do, there's no guarantees that they would accurately reflect the reality of another universe.
We can't even prove that our mathematical axioms accurately describe this universe (in fact we've mathematically proven that such a proof is fundamentally impossible) , how can we hope to prove that they accurately describe all possible universes?
Even with current mathematics we see apparently impossible results, such as the "fact" that you can slice up a single sphere and reassemble the parts into two new spheres, both identical to the first (the Banach-Tarski paradox). That's almost certainly false in this universe, but it's been proven true in mathematics. The implication being that either you *can* duplicate a sphere for free, defying conservation of mass, or at least one of the widely accepted axioms of mathematics is flawed - which would strongly suggest that all the theorems that rely on that axiom are probably also flawed.
There's not even any rule that states that your chosen axioms *should* reflect reality - though they're generally chosen to do so since otherwise the results aren't relevant to this reality. For example - assume you have three infinite lines lying in the same 2D plane: lines A and B are parallel, while line C intersects A. Does C also intersect B? In our universe with Euclidian geometry, yes, always. But choose slightly different axioms and the answer can be no - as is the case in hyperbolic geometry.
And for a long time the field of hyperbolic geometry was considered a novelty with no relevance to the universe we live in. Until we discovered that hyperbolic geometry is actually quite useful for describing Relativity - suggesting that perhaps we actually *don't* live in a universe with Euclidian geometry. And that would imply that all the mathematics based upon the assumption of Euclidean geometry are fundamentally flawed.
(Score: 1) by pTamok on Saturday November 30 2019, @11:06PM (9 children)
No.
Axioms exist separately to the universe.
We choose axioms that generate models that describe our experience of the Universe we inhabit pretty well, but they are 'just' models. That said, there are philosophers who play in the field of the Ontology of Mathematics and Mathematical Platonism who would take the view that if the Universe can successfully be described by mathematics, then it must exist.
There are unsolved problems with our current mathematical model of the universe. Unification of Quantum Theory and General Theory of Relativity is a well known example, but there are others [wikipedia.org].
All interesting mathematical models of the Universe will be flawed in one of two particular senses: they will either be sufficiently complex that Gödels' incompleteness theorems [wikipedia.org] will apply, so it will be possible to make true, but unproveable statements, or they will be so simple that they fail to describe the Universe. There is a possible get-out here: assuming the Universe is finite, then it may be that using mathematics that deals in infinite quantities might not be applicable, but among other things, this means that spacetime must be quantised [wikipedia.org].
Mathematical models are our attempt to rationalise what we see happening around us. There is no guarantee that the Universe is easy to describe, and one of our basic assumptions is that physical laws do not change over time, and especially not discontinuously - but we have no guarantee that gravity will not switch polarity tomorrow - we just have not seen such changes in the past, and so assume they won't happen in the future.
(Score: 2) by Immerman on Sunday December 01 2019, @03:57AM (8 children)
Yes, absolutely.
You can choose axioms with no relationship to the universe, in which case your model will be consistent, but not "true" in any sense relevant to the universe
Or you can choose axioms that, to the best of your understanding, model the actual universe, in which case your model will be as "true" and complete as your axioms - but no more so.
What do you mean that a finite universe would have to be quantized? I've never heard such a claim before, and can't think of any reason it would have to be true. Well, at least not using the usual definition of "inifinite universe" = unbounded system. Plenty of bounded systems are still continuous - the numbers between zero and one for example.
(Score: 2) by legont on Sunday December 01 2019, @04:17AM (6 children)
Wrong twice.
First, you imply scientific truth which has nothing to do with math. Math does not concern with what may or may not be true in say physics. Math is way above this silly idea of being in any relation to the the universe we live in.
Second, just try to come up with such an axiom and see if you can build any nontrivial math. You will fail.
"Wealth is the relentless enemy of understanding" - John Kenneth Galbraith.
(Score: 2) by Immerman on Sunday December 01 2019, @04:33AM (5 children)
Better than that, I'll give you a whole field of mathematics that use axioms that don't reflect reality: non-euclidean geometry. Often only one or two axioms are changed, and the resulting mathematics describe various spatial geometries that are fundamentally incompatible with the Euclidean geometry we're familiar with. They are often very well developed, but the universe they describe is not the universe we live in.
(Score: 2) by legont on Sunday December 01 2019, @04:50AM (4 children)
We sure leave in the universe described by non-euclidean geometry. Does general theory of relativity rings the bell? Gravitational waves made news just recently? They are space geometry waves, mind you.
"Wealth is the relentless enemy of understanding" - John Kenneth Galbraith.
(Score: 2) by Immerman on Sunday December 01 2019, @02:59PM (2 children)
Hyperbolic geometry is indeed useful for describing relativity - but allow me a counterpoint: Draw a line R and a point P. Then draw two distinct lines through point P that don't intersect line R. It's guaranteed that you can do so in hyperbolic space, but guaranteed impossible in Euclidean space.
(Score: 2) by legont on Sunday December 01 2019, @08:03PM (1 child)
Yep, and the point is...?
"Wealth is the relentless enemy of understanding" - John Kenneth Galbraith.
(Score: 2) by Immerman on Tuesday December 03 2019, @02:52PM
Either Euclidean geometry is "false" in this universe, or Hyperbolic geometry is (or both could be). They are mutually exclusive as descriptions of reality.
(Score: 2) by c0lo on Monday December 02 2019, @12:41AM
FTFY.
Because many such geometries exist and all of them except one do not describe the spacetime geometry of the current universe. And yet the are all consistent as mathematical constructs.
https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
(Score: 1) by pTamok on Sunday December 01 2019, @08:19PM
You are correct to question that a bounded universe necessarily requires a discretized/non-contiguous space-time. I was wrong. Sorry.
(Score: 2) by Common Joe on Tuesday December 03 2019, @10:30AM (1 child)
Hi Immerman,
I'm currently taking a math course at the university level. (I did well in math until Calculus. Since then, it's been a challenging.) You seem to have a firmer grip on math than I do. I was hoping to throw a couple of odd questions your way since you seem to be talking about things that pertain to my math class. If you have time, would you be game to answering a few questions for me?
I suggest taking this away from Soylent News so we don't pollute the comment section. Perhaps you could email me? Email Address should be public.
-- Common Joe
(Score: 2) by Immerman on Tuesday December 03 2019, @03:02PM
Sure, you should be getting an email from me shortly, if you haven't already.
(Score: 2) by Immerman on Saturday November 30 2019, @06:15PM (1 child)
No - without thinking being the math theorems wouldn't exist.
Just like Newtonian gravity wouldn't exist without thinking beings. It's a mental model, not a fundamental property of the universe.
>math does not depend on the shape of a universe in question
Doesn't it? If the geometry of the universe were so dramatically different that Box A could be within Box B, while Box B is simultaneously within Box A, then that might fundamentally alter set theory, for starters
And it's not just the geometry of the universe - the fundamental rules might be different in ways we can't even imagine - what if bringing two objects together consistently caused a third object to spontaneously come into existence through similarity resonance or some other nonsense-in-this-unverse. That would alter even basic counting beyond recognition. Without conservation of mass and energy, quantities of things would be perpetually unstable, and there's no reason to believe that all universes must obey such rules. Any mathematics built on the assumption that quantities remain constant except under the influence of mathematical operations would be false.
At its core, mathematics is a set of rules for how you can draw conclusions to build new rules from previous rules, with the new rules being guaranteed to be no less true than the original ones. That aspect might remain - but the inherently unprovable axioms that describe the universe and provide the foundation for the rest might very well have to change
(Score: 2) by legont on Sunday December 01 2019, @04:29AM
Forgive me, bro, but properties of the universe can't alter a given set theory. If they do, you might want to demonstrate the field responsible ;)
Anyway, axiom do not describe the universe. They are simply axioms the universe be damned.
What is amazing though, that an interesting set of axioms always finds something in the universe to describe. The pattern is that once mathematicians describe something, it is later discovered in the universe. Always later, sometimes much later. That's the real mystery.
"Wealth is the relentless enemy of understanding" - John Kenneth Galbraith.
(Score: 2) by maxwell demon on Saturday November 30 2019, @07:13AM (8 children)
But mathematicians don't even agree among themselves about truth.
For example, does the law of the excluded middle hold? Classic logic says yes. Intuitionist logic says no. Who is right?
The Tao of math: The numbers you can count are not the real numbers.
(Score: -1, Flamebait) by Anonymous Coward on Saturday November 30 2019, @07:56AM (5 children)
Well, obviously they cannot both be right, so the one that says they cannot both be right is right, according to the principle of the excluded middle. Oh, non- or as they prefer to call it, -paraconsistent logic is such a hoot!!! But it is like String Theory, in that while internally consistent, it is something like a Runaway1956 of a theory: Big, obese, actually existing, but theoretically impossible. So existentially impossible. There is no Runaway1956, he is only a virtual Soylentil created by Russian bot farms, with intimate knowledge of Arkansaws Hillbilly angst. Poor 'billies! Sucks to vote for a Clinton as a home son!
(Score: 2) by maxwell demon on Saturday November 30 2019, @08:13AM (4 children)
No, that would be the law of non-contradiction. The law of the excluded middle says they can't be both wrong.
The Tao of math: The numbers you can count are not the real numbers.
(Score: 1, Flamebait) by aristarchus on Saturday November 30 2019, @09:36AM (3 children)
Law of excluded middle is naught but the double negation of the law of non-contradiction. But you knew that. Now lets get started on that "law of identity". Why, in God's universe, should A=A? Fichte, and Hegel, were much more comfortable with A=~A, the identity of contradictories. But then both the Excluded Middle and Non-contradiction have to find new homes, do they not?
Ah, this is why I hang out on SoylentNews. That, and bashing alt-right idiots like Runaway and TMB, who could not do metaphysics even if they knew what the word meant.
(Score: 5, Touché) by maxwell demon on Saturday November 30 2019, @10:07AM (2 children)
In God's universe? Well, depends on which God you are thinking of. I mean, "there is no God but God" pretty much relies on the law of identity, doesn't it? On the other hand, there's a well-known God with a son that also is God despite there only being only one God, and that son was God and human at the same time, despite humans not being God. In that God's universe, 3=1, from which it follows that -1=1. Thus negative is positive, which means the devil and God are the same, although they are opposed to each other. So in that God's world the identity of contradictories clearly holds. ;-)
Anyway, on a more serious note, ~(A=A) is not the same as A=~A.
The Tao of math: The numbers you can count are not the real numbers.
(Score: 2) by aristarchus on Saturday November 30 2019, @10:22AM (1 child)
You are a demon, scaring the straights! But please exfoliate on your last assertion
(Score: 2) by maxwell demon on Saturday November 30 2019, @11:18AM
It's simple:
~(A=A): It is not true that A is the same as A. For example, birds are not birds.
A=~A: A is the same as not A. For example, birds are non-birds.
The first doesn't imply the second. If birds are not birds, they still may not be non-birds either.
The second doesn't imply the first. If birds are non-birds, they still may be birds.
The Tao of math: The numbers you can count are not the real numbers.
(Score: 2) by FatPhil on Saturday November 30 2019, @02:48PM
Great minds discuss ideas; average minds discuss events; small minds discuss people; the smallest discuss themselves
(Score: 2) by legont on Saturday November 30 2019, @04:35PM
Mathematitions are not gods and the field is evolving. Your example is simply an evidence that math exists independently from mathematitions and so is objective reality.
"Wealth is the relentless enemy of understanding" - John Kenneth Galbraith.
(Score: 2) by darkfeline on Monday December 02 2019, @12:18AM
That's quite literally wrong. Any system of mathematics is only true in the universes where the axioms from which the system is constructed exist. The set of universes where this is true is strictly a subset of all possible universes. Therefore, no mathematical system is applicable to all universes. QED.
Join the SDF Public Access UNIX System today!
(Score: 3, Informative) by FatPhil on Saturday November 30 2019, @03:47PM (2 children)
Great minds discuss ideas; average minds discuss events; small minds discuss people; the smallest discuss themselves
(Score: 3, Informative) by opinionated_science on Sunday December 01 2019, @12:38AM (1 child)
There you go https://is.muni.cz/el/1441/podzim2013/MA2MP_SMR2/um/Nelsen--Proofs_without_Words.pdf/ [is.muni.cz]Proofs Without Words
A nice browse.
(Score: 2) by FatPhil on Sunday December 01 2019, @01:05AM
Great minds discuss ideas; average minds discuss events; small minds discuss people; the smallest discuss themselves
(Score: 3, Insightful) by aristarchus on Saturday November 30 2019, @12:00AM (10 children)
Now take that sheet of paper, and tape or tack it to the wall. If it was a plastic bag, it would still be there years later, but this is not frojack time. Suddenly the infinity of points on the paper is much smaller than the infinity of points on the wall, since the small part of the wall now contains, or replicates, the paper point, and there is much wall beyond that. So, you're thinking, we need a bigger sheet of paper . . .
(Score: 0) by Anonymous Coward on Saturday November 30 2019, @04:04AM
agree.
(Score: 1, Informative) by Anonymous Coward on Saturday November 30 2019, @06:26AM (7 children)
The fallacy of the summary's quote: infinity is not a quantity.
What bothers me is when people talk about cardinalities. As though there are greater degrees of infinity. It's.. not a quantity -- you don't have a greater degree of "this is sand". It's not the end of a number line, it's not a given number of some "things," it's not a quantity -- it's a concept.
If you want to call cardinal numbers things, then fine -- they're cardinal numbers, but they're not infinity.
(Score: 1) by khallow on Saturday November 30 2019, @07:14AM (2 children)
(Score: 2) by https on Saturday November 30 2019, @04:29PM (1 child)
S/he'd want to do that because the whole point of infinity is that whatever cardinal you can come up with, it won't be big enough. Increasing the amount never does the job. Thus, different.
Offended and laughing about it.
(Score: 1) by khallow on Monday December 02 2019, @06:22AM
For finite cardinals. There's an infinite one that would do the job.
(Score: 2) by maxwell demon on Saturday November 30 2019, @07:17AM
How do you define quantity?
I would say the answer to “how many” is a quantity. And the answer to “how many elements are in that set” may be an infinite cardinal, and it may be a different infinite cardinal for different sets.
The Tao of math: The numbers you can count are not the real numbers.
(Score: 2) by aristarchus on Saturday November 30 2019, @07:19AM
Oh, great, someone who understands! May I suggest to you Ψαμμίτης, [wikipedia.org] a work by the great Archimedes, of whom your may have heard. He mentions yours truly. But bethink thyself, oh modern, that we ancients have thought your problems long before you even thought to think them yourselves.
I have your perturbation of the cosmic constant in my hand. Explain yourself.
(Score: 2) by aristarchus on Saturday November 30 2019, @08:35AM
Very insightful, AC! Infinity is a totality, not a quantity. But this confusion follows mathematics through Cantor. Better to consider Hegel, and earlier philosophers: Infinity is a whole-ness, a one-ness, that than which there is nothing beyond. So the Energizer Bunny idea of infinity, the engineer's idea of infinity, the n+1 idea of infinity, is, shall we say, less than complete.
Freebie philosophy joke: "Why is the Energizer Bunny always in the bathroom?"
Answer: Because he keeps going, and going, and. . . .
(Score: 2) by Immerman on Saturday November 30 2019, @04:35PM
Infinities are not a quantity, and mathematicians will be the first to tell you that. But they do have a "magnitude", and you can see that when you start comparing them.
For example - the limit as x approaches infinity of a(x)=x is infinity. So is the limit of b(x)=x^2.
But the limit of b(x)/a(x) is also infinity, implying that the limit of b(x) is infinitely larger than that of a(x). If those functions described the number of items in two different unbounded sets, then you could conclusively state that for every item in Group A, there are an infinite number of items in Group B. Their cardinalities are fundamentally different.
(Score: 2) by Immerman on Tuesday December 03 2019, @03:36PM
Nope - so long as there's a 1:1 relationship in both directions, the infinities are equal. And any sort of scaling preserves that relationship. For example, there are exactly as many even numbers as there are natural numbers (positive integers), even though you would intuitively think there would be half as many.
Proof:
- For every even number E, there exists a natural number N such that E = 2N - therefore there must be at least as many natural numbers as there are even numbers
- For every natural number N, there exists an even number E such that N = E/2 - therefor there must be at least as many even numbers as there are natural numbers
- If count(all N) >= count(all E) and count(all E) >= count(all N), then count(all N) must be exactly equal to count(all E)
You could write the exact same proof for multiples of a billion instead of 2, and an extremely similar one for the two-dimensional sheet of paper. The paper is just a more visual/intuitive version of the formal proof.
(Score: 4, Disagree) by NickM on Saturday November 30 2019, @12:02AM (11 children)
I a master of typographic, grammatical and miscellaneous errors !
(Score: 0) by Anonymous Coward on Saturday November 30 2019, @12:49AM (2 children)
"There is a minimum distance measurable by an observer (that choice of terminology is quite bad imho) called the Planck distance."
People misunderstand what plank constant attempts to accomplish. It derives universal units that aren't based on arbitrary measurements. For instance trying to say a foot is the length of a specific person's foot or a pound is the weight of this specific piece of metal located at this museum isn't very universal. Whereas Plank attempts to derive units using universal constants. A universal constant is not constrained to an arbitrarily defined unit (like a foot), it can be expressed in different units but the underlying constant is universally the same in value no matter what units you choose to express it in.
"there is no reason to believe that exactly one unit of Planck time has any special physical significance. Rather, the Planck time represents a rough time scale at which quantum gravitational effects are likely to become important."
https://en.wikipedia.org/wiki/Planck_time [wikipedia.org]
(Score: 1, Interesting) by Anonymous Coward on Saturday November 30 2019, @01:22AM (1 child)
To amplify this, the Planck mass is roughly a whole 21µg.
Besides, if you listen to Susskind, he'll tell you the basis unit is area, not length.
(Score: 0) by Anonymous Coward on Monday December 02 2019, @07:01AM
.. except IIRC Susskind then uses Planck Length as the radius of the sphere for which the Planck Area is the surface area, to define the smallest blackhole possible.
(Score: 3, Informative) by legont on Saturday November 30 2019, @01:00AM (1 child)
You describe physics. The so called "light" in the prof above has nothing to do with actual photons or whatever it might be. It is simply a mathematical operation. Physicists do steal them and give them some additional meanings and limitations, but it has nothing to do with the beauty of math.
"Wealth is the relentless enemy of understanding" - John Kenneth Galbraith.
(Score: 2, Informative) by RandomFactor on Saturday November 30 2019, @01:42AM
So he meant ray.
В «Правде» нет известий, в «Известиях» нет правды
(Score: 5, Insightful) by maxwell demon on Saturday November 30 2019, @07:39AM
Well, if you interpret it as a statement about the physical world (as opposed to the classical model of the physical world which we use in everyday's life), you don't have to go to the Planck scale.
To begin with, that “wall” is made out of atoms which are very concentrated cores with electron hulls. It also doesn't have a well-defined surface; the probability of finding an electron bound to an atom of the wall just decreases exponentially with the distance.
Next, there's no such thins as a “beam” of light. And you don't even have to go to quantum mechanics for that; even in classical electrodynamics light is a wave, and you only can approximate it as a beam as long as the size of the beam is large compared to the wave length. Which a point by definition is not.
But then, everything I used in the description above also are models of the world, not the world itself. They are more accurate models in certain respects, but neither is the world itself. That doesn't change if you go to General Relativity or Quantum Field Theory.
Really, all we can talk about really are our models of the world. Which model of the world we use depends on what phenomenon we describe. In principle you could use quantum field theory in curved spacetime in order to describe how an airplane works. Except that in practice you can't because it is far too complex. Therefore you use a simpler model where that complicated arrangement of matter fields is approximated as material with certain properties, the propagation of the field is approximated as movement, and the curved spacetime that field propagates in is approximated as Euclidean space with a homogeneous gravitational field.
Now for some models of reality we have an intuitive understanding because we evolved at a scale where those approximations are valid. And if one of those models happens to fit a certain mathematical concept, there's nothing wrong in using our intuitive understanding of that model to understand the mathematical concept.
The Tao of math: The numbers you can count are not the real numbers.
(Score: 2) by c0lo on Monday December 02 2019, @01:02AM (4 children)
Physicists' truth != mathematicians truth
Mathematicians usually dispense themselves from the relation with the reality as a way to establish the truth values of their theorems.
https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
(Score: 2) by legont on Monday December 02 2019, @04:32AM (3 children)
Actually, this can be phrased quite differently. Currently physics, specifically quantum theory, is rather far from reality in traditional sense. Specifically, they say that reality depends on observer; therefore the reality is subjective.
On the other hand most mathematicians believe that their constructs exist independently of any observer and even hold true in any universe.
There is a recent book - a very good one - that tries to get physics back to realism. I highly recommend it. https://www.amazon.com/Einsteins-Unfinished-Revolution-Search-Quantum/dp/1594206198 [amazon.com]
"Wealth is the relentless enemy of understanding" - John Kenneth Galbraith.
(Score: 3, Interesting) by c0lo on Monday December 02 2019, @07:28AM (1 child)
Nope, it's not quite subjective. The dependency of observer is not specific to quantum mechanics, relativity shows it too - time dilation and whatnot.
However, the dependency of observer still follows rules that allows predictions to be made. That is, the "dependency of observer" != "subjective as in «at the whims of the observer»". No "alternative facts" at it were.
Since I can't accept the premise of "subjective reality" I have a hard time to get what's about in "getting physics back to realism".
I'll try to find some time over the holidays to get and read that book, thanks for the recommendation.
I might not succeed, nowadays I'm gravitating more to things like "the incontestable reality: that cheap table saw is quite shitty, time to rebuild the table for it".
https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
(Score: 3, Interesting) by legont on Monday December 02 2019, @06:24PM
Being a mathematician by training, I am somewhere in the middle between your position and the book's, which I'd call hard realism. Nevertheless, I found it very interesting - enjoy.
"Wealth is the relentless enemy of understanding" - John Kenneth Galbraith.
(Score: 2) by Immerman on Tuesday December 03 2019, @03:58PM
>Specifically, they say that reality depends on observer
Actually, that's specifically the Copenhagen Interpretation of QM (which many consider a cop-out, essentially: don't theorize more closely until we have more data - which was probably reasonable at the time), which states that "things" are purely probability waves until "measured", at which instant they collapse into a particle with a definite state (Heisenberg uncertainty notwithstanding). Neither the Many Worlds Interpretation nor Pilot Wave Theory rely on an observer, to name only two of the more of the many possible interpretations.
Many Worlds avoids it by saying that all possible outcomes do occur, but any particular observer can only see the one that happened in their particular timeline.
Pilot Wave in contrast is completely deterministic - all particles are real particles taking definite paths through space at all times, but are also intrinsically linked to a guiding wave described by the quantum wavefunction, that simultaneously exists at all points in space. (a non-local hidden variable)
Even in the Copenhagen interpretation, reality doesn't actually depend on the observer. The state of a quarticle is indeterminate until measured (no observer required - interacting with a classical system collapses its wavefunction), but the measurement has no control over the outcome, and all observers will see the same outcome.
(Score: 3, Funny) by Bot on Saturday November 30 2019, @12:07AM (24 children)
One of the most elegant IMHO is one of the demonstrations that 0.99999... is equal to 1
- student
- wat
- 0.9999999... is exactly equal to 1
- bullshit
- what do you get when you subtract two equal numbers?
- 0 so wat
- so tell me what is 1-0.999999999....
- it's 0.000... o fuc
Account abandoned.
(Score: 1, Informative) by Anonymous Coward on Saturday November 30 2019, @01:30AM (20 children)
That proof presupposes that 1/inf is zero. If instead it is an imaginary number between zero and the smallest positive real then 0.999...1.
(Score: 0) by Anonymous Coward on Saturday November 30 2019, @01:33AM
0.999...<1
Why does 'Plain Old Text' not escape HTML reserved characters?
(Score: 1) by RandomFactor on Saturday November 30 2019, @01:43AM (5 children)
smallest positive real... you mean the planck real?
В «Правде» нет известий, в «Известиях» нет правды
(Score: 0) by Anonymous Coward on Saturday November 30 2019, @05:22AM (1 child)
I'm not familiar with that term and my googlefu has failed me, but if it is defined as "the smallest magnitude real number greater than zero" then yes.
(Score: 1) by RandomFactor on Saturday November 30 2019, @10:54PM
Real numbers have no quantum constraints like reality appears to, but if they were to have them, what would you call that quanta but the 'planck' real number?
В «Правде» нет известий, в «Известиях» нет правды
(Score: 2) by c0lo on Monday December 02 2019, @01:05AM (2 children)
Fuck your planck, mathematicians need reality solely as a physiological support for their thinking process.
https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
(Score: 1) by RandomFactor on Monday December 02 2019, @01:29AM (1 child)
You might derive splinters [arxiv.org] doing that.
В «Правде» нет известий, в «Известиях» нет правды
(Score: 3, Funny) by c0lo on Monday December 02 2019, @04:45AM
Yeap. No matter, though (only pure thoughts). Still of substance for the topic:
- the source cohomology is coherent (reality can keep its chaos) and any cover is proper (no quid pro quo or other improprieties in those coverups, nothing to leak about, only splint or split)
- derive those splinters or don't derive them, its all the same, 'cause the derived ones are the same as underived splinters
- some results are vanishing on extended scale, hiding behind and beyond ample line bundles
(grin)
https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
(Score: 4, Insightful) by maxwell demon on Saturday November 30 2019, @07:41AM
There is no smallest positive real. For each positive real x, there's the smaller positive real x/10.
The Tao of math: The numbers you can count are not the real numbers.
(Score: 3, Interesting) by khallow on Saturday November 30 2019, @08:10AM (9 children)
Where was any division in that proof? A couple more ways to view this.
10*0.99999... - 9 = 0.99999...
yet there is only one solution to 10*x - 9 = x. And that is x=1. And of course, there's the arithmetic of the initial proof in the first place:
1 - 0.99999... = 0.1 - 0.099999... = 0.01 - 0.0099999... = ... = 0 - 0 = 0.
A key thing missed here is that real numbers are defined precisely to eliminate these boundary cases. For example, two common and equivalent ways to describe real numbers is via slices (also known as Dedekind cuts [wikipedia.org] and Cauchy convergent sequences [wikipedia.org] (or Cauchy sequences for short). In the first case, a slice is simply a set of rational numbers which is closed one way under the usual ordering of numbers. If x belongs to the set and y > x, then y belongs to the set as well. This set is defined to have a unique maximal lower bound. So in the case of 1.000... and 0.9999... there is no rational number between the two. Thus, the set of rational numbers greater than either number are identical and by definition the maximal lower bounds must be the same and the two are the same real number.
The second way to define real numbers as the unique limit of infinite Cauchy sequences of rational numbers. A Cauchy sequence is simply a sequence of numbers with the property that for any small positive number e, we can go along the sequence and eventually find a point where every pair of numbers from the sequence past that point differ by less than e. A common way is the decimal representation of the number. Here, if we consider the two sequences: first, 1, 1.0, 1.00, ... (basically, just 1, 1,1,1,1,...) and the second sequence 0.9, 0.99, 0.999,... We find they're Cauchy. The first is trivial since a constant sequence has pairwise difference of 0 by default. The second has pairwise differences less than 10^-N when you get N or more digits in the sequence. For any e>0, there will be some N>1 such that e > 10^-N and thus, the pairwise difference of numbers N or beyond will always be less than e.
One key property for our purposes, is that if we have an infinite Cauchy sequence, then every infinite subsequence is also Cauchy and also converges to the same real number limit. So make a new sequence where we alternate between elements of the two earlier sequences: 1, 0.9, 1, 0.99, 1, 0.999,....
Now pick some e>0 and N>0 such that e > 10^-N. As before, the two original sequences have pairwise differences less than e once you get past N. But what also happens is that in the new, combined sequence, once you get past the (2N)th element, any pairwise difference of two elements is less than 10^-N and hence, less than e. The only missing pairwise comparisons is between elements of the two sequences. But the first is 1 no matter where you are on the sequence, and the second is 1-10^-M, where M>=N, which gives a difference of 10^-M
At this point, I've provided four such proofs that 0.9999... = 1.000... Now perhaps you're thinking, but what if I don't accept the axioms mentioned above? Well, then you don't have real numbers. Those numbers have been studied, but they lead to freaky consequences. For example, if you treat every decimal sequence as unique (rather than some limit), then you end up with 10-adic numbers. They're a sound mathematical object, but they no longer have the expected properties of the reals. Lots of weirdness comes out. For example, there's no natural 1-1 mapping between 10-adic and 2-adic numbers (what is 0.0100000... and 0.099999... in 2-adic?), but there is between decimal and binary representations of real numbers.
(Score: 1, Flamebait) by aristarchus on Saturday November 30 2019, @08:38AM (1 child)
These are the kind of things I would expect khallow to not understand. Libertariantardiness, and the Vienna Circle, tend to debilitate minds for higher level theory. So, khallow, go find some rich people. They can verifiy your "theories".
(Score: 3, Touché) by khallow on Saturday November 30 2019, @02:14PM
Sorry for not meeting your expectations.
(Score: 2) by maxwell demon on Saturday November 30 2019, @10:20AM (6 children)
Well, strictly speaking, it only shows that if0.99999… corresponds to a real number, then that real number has to be 1.
BTW, you got the p-adic numbers wrong: Those have infinitely many digits to the left, not to the right.
The Tao of math: The numbers you can count are not the real numbers.
(Score: 1) by khallow on Saturday November 30 2019, @02:13PM (5 children)
What is the mathematical definition of "left" and "right"? Clue: any definition of 10-adics which uses powers of 10 can use inverted powers of 10 with equal facility.
(Score: 3, Interesting) by maxwell demon on Saturday November 30 2019, @02:31PM (4 children)
The mathematical definition is "has positive exponent of the base" vs "has negative exponent of the base".
In particular, both in the real number base 10 and in the 10-adic numbers, we have 5+5=10. The 1 here on one digit to the left; it's 10^1, not 10^(-1).
The Tao of math: The numbers you can count are not the real numbers.
(Score: 1) by khallow on Saturday November 30 2019, @02:54PM (3 children)
Do we? I calculate 0 for 10-adic.
5+5=10 in reals.
5+5 = 0 (that is, it is addition modulus 10) in 10-adic (each level of power of ten is its own self-contained ring). You start carrying the ones like in integer math, then you break the 10-adic ring structure.
(Score: 3, Informative) by maxwell demon on Saturday November 30 2019, @03:12PM (2 children)
Thank you for profoundly proving that you have not the slightest clue what you are talking about.
https://en.wikipedia.org/wiki/P-adic_number [wikipedia.org]
The Tao of math: The numbers you can count are not the real numbers.
(Score: 1) by khallow on Saturday November 30 2019, @03:39PM (1 child)
In the next section, they discuss a metric on p-adics which is the inverse of the largest power of p that the two numbers are identical as. That's the same metric as I was mentioning.
Point though to my earlier post is that there's no real distinction between infinite digits to right or left. Right or left is an artifact of the definition. All the consequences of p-adic math hold either way.
(Score: 3, Touché) by maxwell demon on Saturday November 30 2019, @04:54PM
The Dunning-Kruger is strong in you.
The Tao of math: The numbers you can count are not the real numbers.
(Score: -1, Redundant) by khallow on Saturday November 30 2019, @08:12AM
Where was any division in that proof? A couple more ways to view this.
10*0.99999... - 9 = 0.99999...
yet there is only one solution to 10*x - 9 = x. And that is x=1. And of course, there's the arithmetic of the initial proof in the first place:
1 - 0.99999... = 0.1 - 0.099999... = 0.01 - 0.0099999... = ... = 0 - 0 = 0.
A key thing missed here is that real numbers are defined precisely to eliminate these boundary cases. For example, two common and equivalent ways to describe real numbers is via slices (also known as Dedekind cuts [wikipedia.org] and Cauchy convergent sequences [wikipedia.org] (or Cauchy sequences for short). In the first case, a slice is simply a set of rational numbers which is closed one way under the usual ordering of numbers. If x belongs to the set and y > x, then y belongs to the set as well. This set is defined to have a unique maximal lower bound. So in the case of 1.000... and 0.9999... there is no rational number between the two. Thus, the set of rational numbers greater than either number are identical and by definition the maximal lower bounds must be the same and the two are the same real number.
The second way to define real numbers as the unique limit of infinite Cauchy sequences of rational numbers. A Cauchy sequence is simply a sequence of numbers with the property that for any small positive number e, we can go along the sequence and eventually find a point where every pair of numbers from the sequence past that point differ by less than e. A common way is the decimal representation of the number. Here, if we consider the two sequences: first, 1, 1.0, 1.00, ... (basically, just 1, 1,1,1,1,...) and the second sequence 0.9, 0.99, 0.999,... We find they're Cauchy. The first is trivial since a constant sequence has pairwise difference of 0 by default. The second has pairwise differences less than 10^-N when you get N or more digits in the sequence. For any e>0, there will be some N>1 such that e > 10^-N and thus, the pairwise difference of numbers N or beyond will always be less than e.
One key property for our purposes, is that if we have an infinite Cauchy sequence, then every infinite subsequence is also Cauchy and also converges to the same real number limit. So make a new sequence where we alternate between elements of the two earlier sequences: 1, 0.9, 1, 0.99, 1, 0.999,....
Now pick some e>0 and N>0 such that e > 10^-N. As before, the two original sequences have pairwise differences less than e once you get past N. But what also happens is that in the new, combined sequence, once you get past the (2N)th element, any pairwise difference of two elements is less than 10^-N and hence, less than e. The only missing pairwise comparisons is between elements of the two sequences. But the first is 1 no matter where you are on the sequence, and the second is 1-10^-M, where M>=N, which gives a difference of 10^-M<=10^-N<e. Thus, the combined sequence is Cauchy and hence, converges to a unique real number. The two Cauchy subsequences also converge to unique real numbers by definition, and thus, by the observation that infinite subsequences also converge to the same limit, the two subsequences must converge to the same real number.
At this point, I've provided four such proofs that 0.9999... = 1.000... Now perhaps you're thinking, but what if I don't accept the axioms mentioned above? Well, then you don't have real numbers. Those numbers have been studied, but they lead to freaky consequences. For example, if you treat every decimal sequence as unique (rather than some limit), then you end up with 10-adic numbers. They're a sound mathematical object, but they no longer have the expected properties of the reals. Lots of weirdness comes out. For example, there's no natural 1-1 mapping between 10-adic and 2-adic numbers (what is 0.0100000... and 0.099999... in 2-adic?), but there is between decimal and binary representations of real numbers.
(Score: 2) by Bot on Saturday November 30 2019, @04:05PM
Depends, IMHO the proof presupposes you try to subtract by hand, at least using the algorithm they taught in italian schools in the 70s.
1.000..-
0.999..=
now usually you start from the rightmost place, but you can examine the situation on the left side too.
1-0 is 1, UNLESS you need to borrow from the one, in that case it is zero.
Do you need to borrow? sure because in every other place you have 0-9 to compute.
0-9, because of the borrowing would become 10-9, which is 1, UNLESS you had to borrow 1, in that case it is 9-9 which is 0.
Do you need to borrow? sure, because in every other place you have 0-9 to compute.
So as you compute by hand you have 0.000 periodic. which is 0 by definition.
Account abandoned.
(Score: -1, Offtopic) by Anonymous Coward on Saturday November 30 2019, @01:35AM (1 child)
TIL all col kid tlk in 3 ltr wrd.
- wat
- bul
- so wat
- o fuc
(Score: 0) by Anonymous Coward on Saturday November 30 2019, @01:59AM
Not col kid, dmb one.
(Score: 3, Informative) by EETech1 on Saturday November 30 2019, @02:25AM
The way I learned it is:
X equals 0.999999...
So...
10X equals 9.999999...
Subtract
10X - X
That is:
9.999999... minus 0.999999...
The infinite stream of nines cancels itself out so:
9X equals 9
And X equals 1
So 0.999999... equals 1.0
(Score: 2, Disagree) by barbara hudson on Saturday November 30 2019, @12:51AM (49 children)
This "proof" is one of those things that is so bad it;s not even wrong - it simply doesn't deserve any consideration.
Take any two infinities that are equal. Add 1 more element to one of them. You no longer have two equal infinities.
Any attempt to say that they are still equal requires screwing up our definition of "equal." Same as "separate but equal" was also false.
See, I can use examples from the real world as mathematical proof too :-) W00t!
SoylentNews is social media. Says so right in the slogan. Soylentnews is people, not tech.
(Score: 4, Funny) by legont on Saturday November 30 2019, @01:20AM (23 children)
Are you sure you can do it? Let's take an infinity of natural numbers. Please add one and let me know what it is.
"Wealth is the relentless enemy of understanding" - John Kenneth Galbraith.
(Score: 2) by barbara hudson on Saturday November 30 2019, @01:39AM (21 children)
It's Infinity+1. By definition. Same as 100+1 is 100+1. Or 10 apples + 3 peaches is 10 apples + 3 peaches. If I buy 10 apples and 3 peaches, I've bought 10 apples and 3 peaches. That it can also be described less exactly as 13 fruits is irrelevant.
I don't have to calculate what infinity+1 resolves to to say that it's more than infinity by itslef, same as I don't have to resolve what 10 apples and 3 peaches resolves to.
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(Score: 1, Informative) by Anonymous Coward on Saturday November 30 2019, @02:34AM (13 children)
No, it's quite possible to add an element to an infinite set and have the new set be the same size as the old set.
Start with the natural, or counting, numbers set, 1, 2, 3, etc.
Add 0 to the set and you get the whole numbers set. But since it is possible to produce a 1 to 1 mapping of the whole numbers to the natural numbers (y=x+1 where x is a whole number), the sets are the same size.
The integers set is also the same size as the natural set, though the mapping is a bit more complicated.
= x*2, where x is a non-negative integer.
y
= -(x*2)-1, where x is a negative integer.
Infinity is NOT a number, so the arithmetic definition of addition doesn't apply. Read up on set theory to learn more.
(Score: 2) by barbara hudson on Saturday November 30 2019, @02:46AM (9 children)
SoylentNews is social media. Says so right in the slogan. Soylentnews is people, not tech.
(Score: 1, Interesting) by Anonymous Coward on Saturday November 30 2019, @04:23AM (1 child)
By whose definition? Yours? What's your definition, something like: "countable means you can count it"?
So to determine if something is countable you have to factor in: how fast you can count, how many hours per day you can devote to counting and how much longer you will live.
Mathematicians avoid that nonsense by saying a set is countable if you can map the natural numbers onto that set. So instead of pointing at the elements saying "that's one, that's two, that's three ...", you need only specify a function that assigns the numbers that way. So mapping the natural numbers onto themselves by f(n) = n shows that the natural numbers are countable, called a "countably infinite" set.
(Score: 2) by barbara hudson on Monday December 02 2019, @02:07AM
And therein lies the problem. Prove it. Saying it isn't proof. As such, this math cannot be used to solve problems in the real world, such as whether the universe is infinite. That's why we need a definition that can be applied to practical problems. And why we are probably going to have to rewrite our rules repeatedly as we seek to expand our knowledge of the universe. Because the tools we have now, including the current definition of infinity, have given us the wrong answers. And long after we're dead, people will still be deciding to rewrite the basic rules yet again. It won't be infinite rewriting, because the human species genome is relatively unstable ...
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(Score: 3, Informative) by Anonymous Coward on Saturday November 30 2019, @05:19AM
This is just a test message, nothing to see here....
(Score: 0) by Anonymous Coward on Saturday November 30 2019, @05:35AM
From https://mathinsight.org/definition/countably_infinite [mathinsight.org]
This includes the natural numbers, whole numbers, integers, and rationals. This is not possible with the real numbers, so they are uncountably infinite.
(Score: 2) by FatPhil on Saturday November 30 2019, @08:49PM (3 children)
Great minds discuss ideas; average minds discuss events; small minds discuss people; the smallest discuss themselves
(Score: 2) by barbara hudson on Saturday November 30 2019, @10:13PM (2 children)
Or are you claiming that we have a Grand Theory of Everything now where the math works?
It always helps to review your basic assumptions every once in a while. Number theories that don't describe reality need to be thrown out and new theories built.
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(Score: 2) by legont on Sunday December 01 2019, @04:45AM (1 child)
I'll be damned - she is serious.
"Wealth is the relentless enemy of understanding" - John Kenneth Galbraith.
(Score: 2) by barbara hudson on Monday December 02 2019, @03:21AM
SoylentNews is social media. Says so right in the slogan. Soylentnews is people, not tech.
(Score: 2) by c0lo on Monday December 02 2019, @01:18AM
The fact that you totally fail math doesn't diminish your value as a human being.
Thus, you needn't persist in going further on the "wrong" direction.
A set S is countable if there exists an injective function f from S to the natural numbers N [wikipedia.org]
Do not confuse it with "No physical being would be able to count all of them", that's a puny non-sense for mathematicians.
https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
(Score: 2) by barbara hudson on Saturday November 30 2019, @03:20AM (2 children)
Also, for those following along, the use of NaN in programming languages is irrelevant to the discussion. NaN does not represent infinity.
SoylentNews is social media. Says so right in the slogan. Soylentnews is people, not tech.
(Score: 2, Informative) by Anonymous Coward on Saturday November 30 2019, @03:53AM
If there's a one to one mapping between the sets then the sets are the same size, even if they're infinite. Imagine an infinite set of pigeon holes, labeled 1, 2, 3, 4, etc. That's the basic natural numbers set. Put zero into pigeon hole 1, one into pigeon hole 2, etc, and you've mapped the whole numbers onto the natural numbers, showing they are the same size infinity, even though the whole numbers are a superset of the naturals.
Real numbers, on the other, hand cannot be mapped onto the natural numbers, so their infinite set is a different size than the natural numbers' infinite set. Read up on set theory, really. This is all covered there.
(Score: 2) by FatPhil on Saturday November 30 2019, @09:11PM
Mathematicians happily will. All the infinities I know are transfinite numbers. Some are transfinite cardinals, some are transfinite ordinals, but transfinite numbers are, quite explicitly, numbers.
Great minds discuss ideas; average minds discuss events; small minds discuss people; the smallest discuss themselves
(Score: 0) by Anonymous Coward on Saturday November 30 2019, @03:41AM (2 children)
No. Infinity is an idea, not a value. Saying a value is "infinity+1" is akin to saying space is "infinitely huge + 1mm3".
(Score: 2) by maxwell demon on Saturday November 30 2019, @07:52AM
All numbers and all values are ideas.
You won't fine the number one in nature. You might find one apple, but one apple is not the number one. The number one is a concept in our mind with which we describe a property that one apple, one orange, one bird and one human have in common, but one apple and two apples don't.
The Tao of math: The numbers you can count are not the real numbers.
(Score: 3, Insightful) by aristarchus on Saturday November 30 2019, @08:51AM
OMG! Did you just pull a cubic mm out of your ass to add to the universe? Ex nihilo, nihil fit, and that applies to assholes as much as it does to hats that seem to have rabbits come out of them. So, infinity + 1mm3 is still just the same old infinity. If you want to think it is bigger, you are welcome to. Plenty of room for all kinds of wrong ideas here in infinity.
(Or, Crap! Is that a tracking pixel? Google is making the universe expand!)
(Score: 2) by legont on Saturday November 30 2019, @04:19PM (3 children)
Your are wrong about the definition, but it is not what I asked. Give an example please - one number you added to natural numbers and I will prove the result is the same.
"Wealth is the relentless enemy of understanding" - John Kenneth Galbraith.
(Score: 2) by barbara hudson on Saturday November 30 2019, @06:53PM (2 children)
You miss the point - numbers do not have an existence in reality. They are concepts. We can do anything with them we choose because we can manipulate concepts.
Prove that you can add 1 to infinity and still have the same infinity. You can't - because there is only }proof" based on abstractions, arbitrary abstractions. When we talk about an infinite universe (the old steady-state universe) we could certainly conceive of adding a parallel universe to it. Then it would be an infinity twice as big - otherwise the laws of conservation would be violated.
Given that the original Big Bang theory is biting the dust because it doesn't account for almost all the matter and energy in the universe, and we're further away from a Grand Theory of Everything than ever before in history, better be ready to change more of your definitions. Like most theories throughout history, they're probably wrong.
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(Score: 1) by pTamok on Saturday November 30 2019, @11:20PM (1 child)
If concepts are not a part of our reality, what are they? Can you demonstrate the existence of anything other than your thoughts?
(Score: 2) by c0lo on Monday December 02 2019, @01:23AM
Mathematicians don't need to. They only need to say "If you assume this set of axioms as true, then I can demonstrate as true these theorems. Do what you like with them, but if they don't fit your reality, too bad for you".
https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
(Score: 0) by Anonymous Coward on Saturday November 30 2019, @01:42AM
$Infinity+1 Social Security benefits check! I buy stock in Bengay.
(Score: 2) by EETech1 on Saturday November 30 2019, @02:28AM (8 children)
Would you mind looking at my comment above, and see if seems legit?
That's how I learned it...
Cheers!
(Score: 2) by barbara hudson on Saturday November 30 2019, @03:09AM (7 children)
The problem is the numeric representation used, which can only inaccurately express certain numbers with repeating digits. 1/3 ends up being represented inaccurately as 0.333...
3 - 1/3 gives 2-2/3, nothing cancels out in terms of repeating digits.
Look what happens when we use real numbers. 1/3 is (falsely) represented by 0.333... Multiply by 3, you get 0.999..., which is not 1, same as 0.333... is not 1/3. Cancellation of repeating digits after the decimal point is an erroneous artifact of the numerical representation used.
We keep making the mistake of thinking that the thing is not the same as the representation of the thing, same as in OOP. "Ceçi n'est pas un pipe" is a good example. You look at a drawing of a pipe, your first reaction is "of course it's a pipe." But the artist is right, it's not a pipe, just a drawing.
We're so used to symbols representing things that we forget they're only symbols, and cannot accurately represent the real world.
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(Score: 2) by EETech1 on Saturday November 30 2019, @03:34AM (6 children)
If you have infinite digits, the math should work out?
https://en.wikipedia.org/wiki/0.999... [wikipedia.org]
(Score: 0) by Anonymous Coward on Saturday November 30 2019, @06:00AM (4 children)
IIRC, the repeating digits problem is still an open question with proponents on both sides of the argument. The whole thing boils down to whether 1/inf is a real number and therefore zero or an imaginary number and therefore not zero. In the first case 0.3...=1/3 exactly, and in the second 0.3...<1/3.
(Score: 2) by maxwell demon on Saturday November 30 2019, @08:04AM (3 children)
Wrong.
You can define numeric systems in which you can have both infinite numbers and infinitesimal numbers, and have the inverse of an infinite number be an infinitesimal number. But in those systems, 0.99999999… simply does not converge at all. It doesn't converge to 1, obviously, but it also doesn't converge to a number infinitesimally smaller than 1. That's because for any such number, you can go another infinitesimal step down and still are above any real number smaller than 1 [using the mathematical term “real number” here, not making a statement about the ontological status].
The Tao of math: The numbers you can count are not the real numbers.
(Score: 0) by Anonymous Coward on Saturday November 30 2019, @12:44PM (2 children)
The problem with your argument is that you are stopping the addition at some point and claiming that the sum is less than one.
You can't do that because the addition is INFINITE.
I prefer to approach it as recursive addition.
Personally, I find the sum 1/2 + 1/4 + 1/8 + 1/16 ... = 1 the easiest to visualize.
1/2 + 1/2 = 1
We agree on that, right?
1/2 + (1/4 + 1/4) = 1
1/2 + (1/4 + (1/8 + 1/8) = 1
We can continue with this pattern to infinity.
Furthermore, since addition is associative, we can drop the parentheses:
1/2 + 1/4 + 1/8 + 1/16 + ... = 1
(Score: 2) by maxwell demon on Saturday November 30 2019, @02:10PM (1 child)
No, I'm not. The limit of a strictly increasing sequence equals its supremum. What I've showed is that with infinitesimal values, a supremum for this sequence doesn't exist because you always can go an infinitesimally small amount lower and still have an upper bound.
The Tao of math: The numbers you can count are not the real numbers.
(Score: 1, Insightful) by Anonymous Coward on Saturday November 30 2019, @04:14PM
The question is, what is the result of an infinite sum?
In order to have an answer, we must agree on a RIGOROUS DEFINITION OF INFINITY.
Your infinitessimal numbers are one approach; the limit concept is another.
But all these paradoxes and disagreements are the result of using an undefined, vague concept of "infinite."
(Score: 2) by maxwell demon on Saturday November 30 2019, @07:54AM
All digits in the decimal system are finite. But you can have infinitely many digits.
The Tao of math: The numbers you can count are not the real numbers.
(Score: 2) by shortscreen on Saturday November 30 2019, @03:49AM
It's a nice thought experiment. I'd be impressed if a child or somebody from two thousand years ago had come up with it.
I don't know how useful or valid the concept of "number of points in an area" is mathematically, but in terms of physics I find the experiment highly questionable. Every "beam" of light from the wall can't travel through the paper because paper is not entirely transparent and blocks a portion of light. Light also travels from the paper to your eye and from the paper to the wall and in various other directions. From this we should logically assume that multiple beams can originate from a single point, while there's no reason to assume equal numbers of beams from each point. Meanwhile, the paper and wall are at different distances so your eye can't focus on both at the same time. When an image is out of focus, light from one point meets your eye at different points causing a blurry image. By their logic, your eye must have more points than either the paper or the wall...
(Score: 3, Insightful) by khallow on Saturday November 30 2019, @08:21AM (1 child)
I can do the same thing with finite multiples of sets. For example, odds, 2*X+1 map to X and evens, 2*X map to X of a second set of the reals. Thus, I have a 1-1 onto map that maps the natural numbers (and integers in general) to two sets of natural numbers (integers respectively).
(Score: 0) by Anonymous Coward on Saturday November 30 2019, @09:42AM
No, you cannot, khallow. It is above your pay scale. Please place your hands in the orange circles, and wait for Mathematics Control to contact you.
(Score: 2) by FatPhil on Saturday November 30 2019, @02:46PM (4 children)
Great minds discuss ideas; average minds discuss events; small minds discuss people; the smallest discuss themselves
(Score: 2) by barbara hudson on Saturday November 30 2019, @07:17PM (3 children)
And I say the definition is flawed because it is arbitrary, and thought experiments (see my two infinite universes example) show it is flawed.
Saying something is just because doesn't work. The fact that too many people accept logical contradictions when it comes to infinities shows just how fucked up/useless the definition is.
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(Score: 2) by FatPhil on Saturday November 30 2019, @08:24PM (2 children)
Great minds discuss ideas; average minds discuss events; small minds discuss people; the smallest discuss themselves
(Score: 2) by barbara hudson on Saturday November 30 2019, @10:18PM (1 child)
What you are espousing most certainly leads to logical contradictions. We have two choices. Either a mathematics system that says infinity + infinity = infinity (which you side with), that when applied to the universe (where there may or may not be infinities), requires the violation of the laws of conservation, or infinity + infinity = bigger infinity, where the laws of conservation still work.
Dark matter and dark energy changed everything. I didn't like it, but I accept it. Because I can see how ideas need to evolve when faced with new information.
Once half the current mathematicians die off, we'll be able to see new ideas enter the field.
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(Score: 2) by FatPhil on Saturday November 30 2019, @11:38PM
Depends which infinities you're talking about. You're out of your depth, quit digging.
Great minds discuss ideas; average minds discuss events; small minds discuss people; the smallest discuss themselves
(Score: 2) by Immerman on Saturday November 30 2019, @04:46PM (7 children)
No. Infinity is not a number, it's an even more abstract concept,
infinity+1 = infinity. They're the same size. Divide the equation by infinity to see it better:
1 + 1/infinity = 1, or
1 + 0 = 1
Contrast that with infinity * infinity, which is infinitely larger.
(Score: 2) by barbara hudson on Saturday November 30 2019, @06:22PM (6 children)
All numbers are abstractions. See my comment regarding "Ceci n'est pas un pipe". Being abstractions, I can perform any arbitrary operations I want, including adding 1 to infinity. After all, any operations on abstractions are them themselves also abstractions.
The abstraction we use to represent a thing is not the thing. The abstraction we use to represent infinity is not itself infinity. The abstraction we use to add 1 to any abstraction, including another number, is itself an abstraction, as are the numbers involved.
Numbers are abstract; we use them to represent things, but they are not those things. 1+1=2 is an abstraction. I rabbit + 1 rabbit = 20 rabbits shows how careful we have to be when dealing with abstractions as if they were real things.
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(Score: 2) by Immerman on Saturday November 30 2019, @06:59PM (5 children)
Yes, numbers are an abstraction, but they're an abstraction of a quantitity
Infinity is NOT a quantity, it's an further abstraction of a conceptual trajectory of quantities with no physical meaning. It can in some ways be algorithmically treated as it it were a quantity, but only if you're very careful to avoid any situation where its true nature would render the algorithm meaningless. e.g.
1/infinity=0
7/infinity=0
therefore 1/infinity=7/infinity
therefore 1=7
The initial claims are true, and everything I did was valid math - but the fact that it was performed with the non-quantity infinity allowed me to prove a false equivalency
(Score: 2) by barbara hudson on Saturday November 30 2019, @07:23PM (4 children)
Two math operations that give the same result do not mean that the original operands are equal.
1 cat + 1 dog = 2 animals.
1 rat + 1 dog = 2 animals.
But the cat is not the same as the rat.
You start with the assumption that all infinities are equal, then conclude that they are all equal. That's what your math example does, and it proves nothing because it assumes that which it attempts to prove.
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(Score: 2) by Immerman on Saturday November 30 2019, @09:25PM (3 children)
You are right that in my proof I assumed that the infinities are equal, which is what allowed me to cancel them out. But I never made any assumption that *all* infinities are equal, just that I was using the same infinity in both initial statements.
My math holds perfectly, only my initial assertions fail. It can be proven that 1/x is not equal to zero for any real number x. The fact that it *does* equal zero for infinity (*any* infinity) allows me to "prove" a falsehood - not because I'm making any assumptions about infinity, but because I'm implicitly making a false claim by asserting that 1/infinity=0 is a valid mathematical statement, rather than an expression of a conceptual abstraction beyond the realm of mathematics to address.
In general the rule of thumb when doing math involving infinities is, don't. Not without a lot of experience. You can get away with it when working with limits (including integrals), but only because you are very careful to never look directly at the infinity, only at the behavior of the function as you get close to it. The instant you acknowledge that you're working with infinity, mathematics ceases to be relevant.
(Score: 2) by barbara hudson on Saturday November 30 2019, @10:07PM (2 children)
Just a thought.
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(Score: 2) by Immerman on Saturday November 30 2019, @11:08PM (1 child)
Why? We have zero evidence that anything real is infinite, the universe included. Even the number of universes that have coalesced out of the inflationary multiverse will be finite, assuming that the inflationary period actually had a beginning and has thus been happening for a finite length of time.
That said, there are fields of mathematics that deal with infinities, carefully, and you can add infinities to other infinities, as well as multiply them, etc. You just can't add *quantities* to infinities and expect anything to happen. Any quantity, no matter how large, is exactly 0% of the size of infinity, so adding it to an infinity has no effect.
(Score: 2) by barbara hudson on Saturday November 30 2019, @11:23PM
We also have zero evidence that the universe is not infinite, and texts until the 60s mostly talked about the steady-state universe that is infinite, has always existed, and always will.
I'm not saying it's the case - quite the contrary. Just that we now know that we know less than we thought we did even a decade or two ago, and that we should think outside the box.
At the turn of the century we were supposed to be just a decade from a Grand Theory of Everything. Now it's worse than nuclear fusion, which has been 20 years in the future for the last 50 years.
It may also turn out that the question is unanswerable.
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