Naked Science Forum
General Science => General Science => Topic started by: talanum1 on 03/12/2020 16:09:10
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One has:
m∞ = n∞ for any ∞ > m, n ∈ Real numbers, since ∞ is the biggest number. Now we can divide through by infinity to get:
m = n for any n, m.
Doesn't this falsify Mathematics?
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since ∞ is the biggest number.
No, it is defined as greater than the largest number.
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No, you have successfully falsified your own misconception.
There are some holes in mathematics (many involving infinities), but this is not one of them.
https://en.wikipedia.org/wiki/P_versus_NP_problem
https://en.wikipedia.org/wiki/Axiom_of_choice
https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
(I don't understand 90% of this, but it makes for interesting reading)
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No, it is defined as greater than the largest number.
That makes no sense: there is nothing larger than the largest number.
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No, it is defined as greater than the largest number.
That makes no sense: there is nothing larger than the largest number.
That's OK, infinity is an abstract concept.
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Now we can divide through by infinity
No, we can't just divide by infinity, and hope to get sensible answers (any more than we can divide by zero and hope to get a sensible answer).
This "divide by infinity" problem frequently comes up in mathematics, and the way mathematicians often get around it is by using the idea of "Limits".
- Instead of saying "infinity", say "take a large number x". Now what happens as x approaches infinity?
- This is written as Lim (for Limit) as x→∞.
- The same trick often works for dividing by 0
Now let's take your example (adjusted a bit to read better):
m∞ = n∞ for any n > m; n, m ∈ Real numbers
- instead of infinity, take a large number x.
- Now what happens as x →∞?
- nx > mx for all values of x (since n>m)
- Now we can divide through by x to get:
- n > m
- This applies for any n, m satisfying n > m, and applies for any value of x, even x→∞.
This proves mathematics (if you do it properly).
Limits are extremely useful for many common mathematical operations like integration and differentiation.
- These produce important relations like Kinetic Energy E=½mv2 (where the ½ and the squared are related)
- And they involve getting sensible answers from limits that effectively approach 0 x ∞
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No, it is defined as greater than the largest number.
That makes no sense: there is nothing larger than the largest number.
Let N be the largest real number.
What is N+1? 2N? N^2? N/0.3? These are all valid operations on any real number. However large you make N, you can always define and manipulate a larger number N'. So ∞ must be larger than the largest number if it has any meaning.
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there is nothing larger than the largest number.
There is no largest number.
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(I don't understand 90% of this, but it makes for interesting reading)
I wish posters on this site were allowed to put "mottos" under their posts. Like they can on other sites.
If they were, your brilliant aphorism: "I don't understand 90% of this, but it makes for interesting reading", would be perfect!
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There is no largest number.
https://www.smbc-comics.com/comic/the-largest-number
https://www.smbc-comics.com/comic/the-largest-number-2
I wish posters on this site were allowed to put "mottos" under their posts. Like they can on other sites.
members->forum profile
which gets you to:
https://www.thenakedscientists.com/forum/index.php?action=profile;area=forumprofile
Put your motto under the signature section, just above the map.
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There is no largest number.
I agree. Say there is a largest number n. Then no matter how large you make n, you can always define n + 1. This shows there is no largest number, hence no infinity.
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I
Put your motto under the signature section, just above the map.
Er, I may be dumb, but what "map"?
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I agree. Say there is a largest number n. Then no matter how large you make n, you can always define n + 1. This shows there is no largest number, hence no infinity.
Non-sequiter
The logic follows only if infinity is defined as the largest number (as you have in the OP), or something greater than such a nonexistent number (As Alan has in the 1st reply).
Perhaps it would be best to use the actual definition instead of all the nonsense in this thread. Infinite is an adjective which translates roughly to 'without bound'. Infinitiy is something like 'unbounded' and isn't a number with which one can meaningfully do arithmetic. Hence most of the logic put forth is meaningless, and mathematics is safely not falsified.
Er, I may be dumb, but what "map"?
There's a map on that page to allow you to indicate where you're from if you so choose. The signature section is just above it. Scroll down a ways, since the page does not fit on a screen.
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There is no largest number.
I agree. Say there is a largest number n. Then no matter how large you make n, you can always define n + 1. This shows there is no largest number, hence no infinity.
Although I agree with your mathematical idea, could there be an actual physical " largest number"?
For example, suppose you could count the numbers of electrons in the Universe. That might turn out to be, say 99 billion trillion trillion trillion. Or whatever.
Whatever it is, it's the largest number you can physically count. You've counted every single electron in existence, throughout the entire Universe. Wouldn't that be the largest "real, physical, number"?
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When you have counted the N electrons, you can count the protons. I'll save you the trouble - it's N again, so we now have a number twice as large as your largest physical count, and we still haven't considered photons, which are being produced all the time. Or neutrons. Or quarks.
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count the numbers of electrons in the Universe. That might turn out to be, say 99 billion trillion trillion trillion
The current estimate is around 1080 (give or take a factor of maybe a thousand).
- There are more neutrinos.
- There are far more (hypothetical) gravitons
- And physicists are just guessing when it comes to the number of Dark Matter particles.
See: https://en.wikipedia.org/wiki/Eddington_number
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When you have counted the N electrons, you can count the protons. I'll save you the trouble - it's N again, so we now have a number twice as large as your largest physical count, and we still haven't considered photons, which are being produced all the time. Or neutrons. Or quarks.
Isn't this mere quibbling. About photons, quarks and stuff. The practical point is this:
The Universe is everything that exists. There can't be anything outside the Universe. So the Universe is a finite thing.
A finite thing can only contain a finite amount of information. So when the information is written in numbers, there will only be a finite amount of actual physical numbers to write it in.
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There is no largest number.
I agree. Say there is a largest number n. Then no matter how large you make n, you can always define n + 1. This shows there is no largest number, hence no infinity.
Unless, of course, infinity isn't actually a number.
If it isn't, there's no problem.
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There is no largest number.
I agree. Say there is a largest number n. Then no matter how large you make n, you can always define n + 1. This shows there is no largest number, hence no infinity.
Unless, of course, infinity isn't actually a number.
If it isn't, there's no problem.
You have hit the nail on the head. "Infinity" isn't a number. Nor does it represent a physical location, which we could reach if we travelled far enough. There's no such thing as "infinity". It doesn't exist.
The only reason we think it might exist, is purely linguistic. "Infinity" is a noun. And a noun is supposed to represent a "Thing". Therefore we're lead to think that "infinity" must be a "Thing".
But suppose we replace the noun "Infinity" by a functionally equivalent adverb: "endlessly".
So instead of saying: "Space extends to Infinity", we say "Space goes on endlessly".
Then the concept of "Infinity" as a thing, disappears.
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The current estimate is around 1080 (give or take a factor of maybe a thousand)
This might be an estimate for the visible universe, a finite volume. The visible universe is not all that exists by at least some definitions of existence.
To say that there is some finite number of Electrons in the universe seems contradictory to most models currently accepted (per cosmological principle) that don't support some kind of edge to it, like the flat Earther's have.
The Universe is everything that exists. There can't be anything outside the Universe. So the Universe is a finite thing.
Non-sequitur. The universe is quite capable of being unbounded in size and still being everything that exists.
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There can't be anything outside the Universe. So the Universe is a finite thing.
sadly, an irrelevant non sequitur.
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Is Mathematics Falsified?
It depends on your assumptions.
Unlike physics, mathematics can operate in a purely hypothetical environment. You may say:
(1) 10+3 = 13
(2) 10+3 = 1
And both statements may be true, if you know that:
(1) applies to the natural numbers
(2) applies to a clock face (which is a modification of modulo arithmetic)
Within a particular framework of assumptions, mathematicians strive for:
- Consistency: Eg you can't prove 1=2
- Completeness: To be able to identify whether any particular statement is true or false.
Unfortunately, in 1931, the mathematician Gödel proved mathematically that you can't prove everything mathematically.
- Any (sufficiently complex) mathematical framework cannot be both consistent and complete.
See: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
In practice, when mathematicians get "stuck" in proving a statement, they often appeal to a proof in a "larger" context
- So to prove something within a particular mathematical framework, they sometimes reach outside that mathematical framework for help...
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Infinite is an adjective which translates roughly to 'without bound'.
Then infinity means ±∞.
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Depends which way you are going. The term "semi-infinite" turns up in a lot of applied maths, physics, and electrical engineering, meaning "starting from zero and fading into the far distance"
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Dividing an infinite number by infinity gives a result that is undefined; it could be any number at all, so you haven't accounted for the infinite number of other possible solutions to both side of your equation, and that is why you have an impossible result.
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It is:
lim_(x -> ∞) x/x = 1
where the same number goes into both x's, and on both sides of the other equation (in the opening post) the same number is substituted. I saw a video where division by infinity is allowed:
√8/∞ = 0.
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You have to be really careful taking things to an infinite limit. There's an infinite number of ways to 'go to infinity':
lim_(x -> ∞) nx/x = n
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There is a one to one pairing in the infinities of the left side and right side. Whatever infinity gets substituted in LS also goes into RS.
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You can only do a 1-to-1 pairing of "countable" infinities. If you venture into "uncountable" infinities, that doesn't work anymore.
For instance, it is just as reasonable to set up a pairing of all the numbers on the continuous Real number line with all of Real numbers on the continuous interval between any two Real numbers. (ie there are just as many Real numbers between 1 and 2 as there are between 1 and 200,000,000) go figure!
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You can only do a 1-to-1 pairing of "countable" infinities.
Specify the four infinities as countable then (and correlated). The range of values that can substitute can be specified as countable, can't it?
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You have to be really careful taking things to an infinite limit. There's an infinite number of ways to 'go to infinity': lim_(x -> ∞) nx/x = n
Yes, you do need to take care with limits.
- But in this example, you don't need much care.
- This equality is true for all values of x≠0, whether x is any real value (or any complex value), any fraction, any integer, any prime number or any Mersenne prime number.
- It applies whether there is a countable or uncountable number of x values tested.
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Mathematics is not falsified if we assume ∞ is a foggy range of values so that ∞/∞ != 1.
But then for a countable ∞ we have ∞ != ∞.
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There is no shortage of apparent contradictions and paradoxes when dealing with infinity. Many times these can be resolved by making sure that the question and context are properly defined.
For instance, with no context, ∞ – ∞/2 is undefinable. But if we give some context, the answer can be defined in a way that is both meaningful and unambiguously provable to be true.
Imagine a farmer planting a row of seeds, and never stopping. There is a crow that follows behind him and eats the seeds at precisely half the rate that the farmer is planting them.
At any given finite value of time, there will be a non-zero number of seeds in the ground. But if we want to know what the "final" result is, we run into problems without knowing more.
If the crow eats every other seed (let's say it eats the odd ones: 1, 3, 5, 7, 9....), then there will be infinitely many seeds in the ground after infinite time passes, and we can say exactly where they will all be (holes 2, 4, 6, 8, 10....). For any hole, if you know whether it is even or odd, you will know whether there is a seed in it or not.
On the other hand, if the crow eats each seed in sequence (at half the rate that the farmer plants them), then there will be no seeds that can be found in the ground. How can I prove this? Pick a hole, any hole. Did the farmer put a seed in it? yes, at time = t. Did the crow eat the seed? yes, at time = 2t. So there are no holes that you can find a seed in.
Is this useful? It can be if you ask the right questions.
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if we assume ∞ is a foggy range of values so that ∞/∞ != 1
I agree that "a foggy range of values" might be a helpful way of thinking about it.
But if it is a foggy range of values, then "∞/∞ != 1" is trying to define a specific value in this foggy range that ∞ cannot be. That doesn't sound very foggy to me.
In fact, if we take the previous example of limits: lim_(x -> ∞) nx/x = n
- Now, if you set n=1, you have the example lim_(x -> ∞) x/x = 1
- This implies, in this particular limit, ∞/∞ = 1, which violates your assertion
So, "a foggy range of values" may be a helpfully foggy concept, but don't try to put precise values on what it isn't!
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if we assume ∞ is a foggy range of values so that ∞/∞ != 1
I agree that "a foggy range of values" might be a helpful way of thinking about it.
But if it is a foggy range of values, then "∞/∞ != 1" is trying to define a specific value in this foggy range that ∞ cannot be. That doesn't sound very foggy to me.
In fact, if we take the previous example of limits: lim_(x -> ∞) nx/x = n
- Now, if you set n=1, you have the example lim_(x -> ∞) x/x = 1
- This implies, in this particular limit, ∞/∞ = 1, which violates your assertion
So, "a foggy range of values" may be a helpfully foggy concept, but don't try to put precise values on what it isn't!
I wonder whether this "foggy range of values" arises from the use of base-10 decimal arithmetic. This arithmetic allows a degree of "fogginess".
For example, the difference between our numerals "7", "8" and "9" is quite small. Their values are close.
Whereas in base-2 binary arithmetic, the difference the numerals "0" and "1" is precise and clear.
Does that mean that we should abandon decimals, and start doing maths in Binary Notation?
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There is no number that can be expressed in base 2, which also cannot be expressed in base 10. There are some inherent mathematical advantages to using base e, but really the only special thing about base 2 is that it is easiest to compute using base two and two-state 0/1 switches.
But there is no difference between any base when discussing infinity.
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Does that mean that we should abandon decimals, and start doing maths in Binary Notation?
Most computers already do. Simple calculators may still be using decimal arithmetic.
I agree that teaching binary multiplication tables is much easier than teaching decimal, and lets you focus on the essentials:
0x0 = 0
0x1 = 0
1x0 = 0
1x1 = 1
But there is no need to abandon Decimal arithmetic because it is "inaccurate" or "foggy" in some sense.
- There is an exact 1:1 mapping from every decimal integer to its binary counterpart.
the difference between our numerals "7", "8" and "9" is quite small. Their values are close.
What you are describing here is that many quantities in this world span a great range of values, and it is easier to make sense of them as ratios. So the ratio 2/1 is much bigger than 9/8.
A number of our senses work in this way; the loudness of sounds is represented on a logarithmic scale called "decibels".
See: https://en.wikipedia.org/wiki/Decibel
Many other systems in the real world work this way, even accounting records for a company.This has been used to detect when a fraudster has tried to hide their trail.
See: https://en.wikipedia.org/wiki/Benford%27s_law
Oops! Overlap with chiralSPO...
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Mathematics is not falsified if we assume ∞ is a foggy range of values so that ∞/∞ != 1.
(a) You haven't demonstrated mathematics to be falsified under any conditions
(b) You shouldn't use ∞ to denote a number x such that x/x! = 1 because ∞ is already reserved and understood for other purposes.
(c) a range of values, foggy or otherwise, implies that you could write down any real number somewhere in that range. But since ∞ is defined as greater than any number you can think of, x ≠ ∞
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if we assume ∞ is a foggy range of values
The idea of Limits is to provide more information, pinning down a "foggy range of values" to a more precise range of values, and sometimes, to a single value.
Sometimes the Limit is still infinite, but in many useful cases, the limit is finite, and a fixed value that you can use.
- So Limits provide more information than you get by just assuming that "∞ is a foggy range of values"