Do Infinite Pencils Exist?(mitchgordon.me)
mitchgordon.me
Do Infinite Pencils Exist?
http://mitchgordon.me/math/2022/02/01/infinity.html
19 comments
Infinity does not have to be a "repeatable process that we can keep doing forever." That seems like a myopic view.
Anyways, their reasoning in the rest of the article only holds if you assume the universe is finitely bounded in time and space.
Anyways, their reasoning in the rest of the article only holds if you assume the universe is finitely bounded in time and space.
I'm not convinced their reasoning holds for the rest of the article even if the universe is finitely bounded in time and space. Recall only takes one counterexample to refute.
Take the example given to us by Heraclitus, "You cannot step into the same river twice, for other waters are continually flowing on." My claim extends it such that any river is bundle of infinitely many unique river states. It rests on the infinite divisibility of space. Remember that although the real interval [0, 1] is a finite bounds, there are an infinite number of reals which lie inside it.
Take the state of a river at any one point in time. Select any one molecule of water. A finite spatial bounded interval anchored at one end by the selected molecule and the other by a small and arbitrary distance away, say 1cm. It is possible to construct new river states by repositioning the molecule of water along the infinitely possible positions along the spatial interval.
Now, realize that this is a special case of a river simply and naturally flowing as all water molecules are interpolatable as they flow. Thus as rivers flow they occupy an infinite number of continuously possible states.
So no, I don't think finite bounds of time and space are sufficient. You would also need to have discretized space and time and that's so far not been confirmed.
Take the example given to us by Heraclitus, "You cannot step into the same river twice, for other waters are continually flowing on." My claim extends it such that any river is bundle of infinitely many unique river states. It rests on the infinite divisibility of space. Remember that although the real interval [0, 1] is a finite bounds, there are an infinite number of reals which lie inside it.
Take the state of a river at any one point in time. Select any one molecule of water. A finite spatial bounded interval anchored at one end by the selected molecule and the other by a small and arbitrary distance away, say 1cm. It is possible to construct new river states by repositioning the molecule of water along the infinitely possible positions along the spatial interval.
Now, realize that this is a special case of a river simply and naturally flowing as all water molecules are interpolatable as they flow. Thus as rivers flow they occupy an infinite number of continuously possible states.
So no, I don't think finite bounds of time and space are sufficient. You would also need to have discretized space and time and that's so far not been confirmed.
This post mirrors the the medieval take on infinity.
Modern mathematics is happy with actual infinities.
Modern mathematics is happy with actual infinities.
Yeah, this was a VERY controversial subject for a long time. Maybe even still today!
David Foster Wallace's Everything and More was an enjoyable history of the topic.
I think the rebuttal to the author's "write all the primes down, you can't" would be something like "tell me how many numbers there are between 0 and 1." I'd die before I'd finish writing out all the primes, but the author or that professor referenced in the article could just never even have a way to give me an answer to my request. (EDIT: If they say that's not "real-world" enough, how many possible pieces of wood of distinct length could they cut that would be between 0 and 1 meters long? Unless someone wanted to try to brute-force it based on like molecule size... maybe it's not truly infinite? But could you even measure that difference? Which brings you back to infinity from a precision aspect...)
David Foster Wallace's Everything and More was an enjoyable history of the topic.
I think the rebuttal to the author's "write all the primes down, you can't" would be something like "tell me how many numbers there are between 0 and 1." I'd die before I'd finish writing out all the primes, but the author or that professor referenced in the article could just never even have a way to give me an answer to my request. (EDIT: If they say that's not "real-world" enough, how many possible pieces of wood of distinct length could they cut that would be between 0 and 1 meters long? Unless someone wanted to try to brute-force it based on like molecule size... maybe it's not truly infinite? But could you even measure that difference? Which brings you back to infinity from a precision aspect...)
That's easy, one piece of wood! A right triangle 1 meter long
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I think this depends on if you're a constructivist, no?
This is silly.
Numbers are concepts, not concrete things. In the same way that authors of autobiographies are not literally held within the pages of the books they write about themselves, the numerals you write on a page are a representation of the actual thing, not the thing itself.
Numbers are concepts, not concrete things. In the same way that authors of autobiographies are not literally held within the pages of the books they write about themselves, the numerals you write on a page are a representation of the actual thing, not the thing itself.
This is a controversial topic within mathematics.
https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Math...
https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Math...
> controversial topic within mathematics
Yes, you are certainly right, within the group of people who think about the philosophy of math. But that is what makes OP's:
> When a mathematician says infinity they mean a repeatable process that we can keep doing forever.
obviously false. Some philosophically-minded mathematicians may well agree with that, but certainly not all. And as for the "average" professional mathematician who never thinks about that stuff, they would not at all agree with OP.
Yes, you are certainly right, within the group of people who think about the philosophy of math. But that is what makes OP's:
> When a mathematician says infinity they mean a repeatable process that we can keep doing forever.
obviously false. Some philosophically-minded mathematicians may well agree with that, but certainly not all. And as for the "average" professional mathematician who never thinks about that stuff, they would not at all agree with OP.
There can be infinite numbers, but there cannot be infinite paperclips.
There are roughly 3,000 sexdecillion paperclips in the universe before you run out of universe and consume your paperclip-making systems into paperclips.
https://www.decisionproblem.com/paperclips/index2.html
There are roughly 3,000 sexdecillion paperclips in the universe before you run out of universe and consume your paperclip-making systems into paperclips.
https://www.decisionproblem.com/paperclips/index2.html
I think the author might be conflating sizes of sets and "calculating" with infinity as if it were a number.
Sets really can have a cardinality of infinity (and beyond). If I have any number of primes, I can construct another prime which is different from all primes I've had before. That makes the total number of primes infinite.
On the other hand, you have things like infinite sums, infinite fractions, irrational numbers, etc which appear to require an infinite series of steps to calculate and nevertheless yield a result that we somehow know. That can appear paradoxical and discouraging when learning. I think in those situations, it can help a lot to remember we're really talking about limits: An infinite sum is not an actual sum - rather it's a set of sums which differ only in one parameter, the number of parts that are summed. Each individual sum is perfectly ordinary, but if you put in ever-larger values for the parameter, the results of the individual sums might get ever closer to some particular number - the "result" of the infinite sum.
Sets really can have a cardinality of infinity (and beyond). If I have any number of primes, I can construct another prime which is different from all primes I've had before. That makes the total number of primes infinite.
On the other hand, you have things like infinite sums, infinite fractions, irrational numbers, etc which appear to require an infinite series of steps to calculate and nevertheless yield a result that we somehow know. That can appear paradoxical and discouraging when learning. I think in those situations, it can help a lot to remember we're really talking about limits: An infinite sum is not an actual sum - rather it's a set of sums which differ only in one parameter, the number of parts that are summed. Each individual sum is perfectly ordinary, but if you put in ever-larger values for the parameter, the results of the individual sums might get ever closer to some particular number - the "result" of the infinite sum.
Euclid’s proof is famously not constructive, so it doesn’t directly give us a method for constructing a new prime.
But it is. He says that given a list of primes P1, P2, ..., PN, construct (P1 * P2 * ... * PN) + 1, which is a new prime not on the list, so given any finite list we can always construct more primes.
But it is. He says that given a list of primes P1, P2, ..., PN, construct (P1 * P2 * ... * PN) + 1, which is a new prime not on the list, so given any finite list we can always construct more primes.
No, the constructed number (P1 * P2 * ... * PN) + 1 either is prime or has a prime factor greater than PN.
For example, (2 * 3 * 5 * 7 * 11 * 13) + 1 is 30031, which is 59 * 509
(Apparently I misremembered the details of Euclid's proof. See the comments below.)
For example, (2 * 3 * 5 * 7 * 11 * 13) + 1 is 30031, which is 59 * 509
(Apparently I misremembered the details of Euclid's proof. See the comments below.)
It's not necessarily a prime bigger than PN as the original proof is about some list of primes, not all primes less than some prime. E.g. 3,5 becomes 16, which is missing 2.
Apparently I misremembered the details of the proof.
I thought that it started with (P1, P2, P3, ..., PN) being a list of all the primes from 2 up to PN. Since (P1 * P2 * ... * PN) + 1 either is prime or has a prime factor not in the list, that prime factor must be greater than PN.
Apparently (according to Wikipedia's summary) Euclid started with an arbitrary list of primes, and showed that for any such list there is a prime not in the list.
Either method works to show that there are infinitely many primes.
I thought that it started with (P1, P2, P3, ..., PN) being a list of all the primes from 2 up to PN. Since (P1 * P2 * ... * PN) + 1 either is prime or has a prime factor not in the list, that prime factor must be greater than PN.
Apparently (according to Wikipedia's summary) Euclid started with an arbitrary list of primes, and showed that for any such list there is a prime not in the list.
Either method works to show that there are infinitely many primes.
Yes, I meant to say "given a purported list of all of the primes".
Euclid's proof constructs a number which is not a multiple of the purported primes, but it does not construct a new prime.
That's technically true of the exact version of the proof usually presented, but it's a really bad example of a nonconstructive proof, because it can trivially be made constructive by taking the least integer greater than PN that is a factor of (P1*...*PN)+1 as your new prime.
It constructs a number that is not divisible by any member of the list, so either it is prime, or there is some other prime that is not a member of the list. Either way we can grow the list.
Because the list you start with isn't actually a list of all the primes, the number it comes up with isn't necessarily prime.
The product of primes plus one isn't always prime.
For example: 3*5 + 1 = 16, not prime.
Euclid's argument is that the product of some primes plus one is either prime or not; if it's prime, then it's an example of a prime that is not in the list. If it is not prime, then Euclid argues that the number's prime factors are not in the list.
To directly use Euclid's result to find a new prime number, we have to multiply some existing numbers together, add 1, and then factor that number. If it doesn't factor, then it is a new prime; otherwise its factors are new primes.
That seems constructive; all of the steps amount to a terminating algorithm that will give us at least one new prime.
For example: 3*5 + 1 = 16, not prime.
Euclid's argument is that the product of some primes plus one is either prime or not; if it's prime, then it's an example of a prime that is not in the list. If it is not prime, then Euclid argues that the number's prime factors are not in the list.
To directly use Euclid's result to find a new prime number, we have to multiply some existing numbers together, add 1, and then factor that number. If it doesn't factor, then it is a new prime; otherwise its factors are new primes.
That seems constructive; all of the steps amount to a terminating algorithm that will give us at least one new prime.
I think you are missing a step in the proof. That could still be a composite, see here: https://newbedev.com/understanding-euclid-s-proof-that-the-n...
The article is just as dumb as the anecdote it describes. He even realizes it at the end.
So what if the computer runs out of memory, or you can't write them all out. Just because I can't count to a googol[1] in my head doesn't mean it doesn't exist.
[1]: https://en.wikipedia.org/wiki/Googol
[1]: https://en.wikipedia.org/wiki/Googol
In computation, we start with some information and then get other information from it, but the operations to get there have certain time and space complexity.
Math works the same except every operation is O(0) time and space complexity. To a mathematician, you start with set of some axioms and then -- boom -- everything possible thing you could derive from that is said to exist.
There's a place for both. A mathematician needs to be concerned with what things are true, not how hard it is to see it. But someone who actually needs (or has to manage) specific information may need to be concerned with the difficulty of that.
Math works the same except every operation is O(0) time and space complexity. To a mathematician, you start with set of some axioms and then -- boom -- everything possible thing you could derive from that is said to exist.
There's a place for both. A mathematician needs to be concerned with what things are true, not how hard it is to see it. But someone who actually needs (or has to manage) specific information may need to be concerned with the difficulty of that.
That is, they mistakenly believe that infinity actually exists. Like, in the real world.
Well, for all we know, they universe might be infinite and contain an infinite number of galaxies, stars, atoms, ...
Well, for all we know, they universe might be infinite and contain an infinite number of galaxies, stars, atoms, ...
The universe most certainly is not infinite by any measure and does not contain infinitudes.
You seem very sure for such a wild claim. The observable universe is finite, as for the whole universe, we have no idea (unless you know something that I don't).
But we have at least measured the curvature of the observable universe and within experimental error the result is that it is flat. Which in turn would imply an infinite universe if this flatness is exact and general, i.e. not something exceptional in the observable universe.
Do we know that for certain? We think space is still continuous right? So there's an infinity between any two points.
Even with all the plank-length stuff, they're not snapped to a plank grid right?
I'm genuinely asking: my level of knowledge is YouTube.
Even with all the plank-length stuff, they're not snapped to a plank grid right?
I'm genuinely asking: my level of knowledge is YouTube.
Yes actually, particle motion is quantized on the planck scale, that is, snapped to the grid. That's why it's called quantum physics. Modern theories even have it that there are no particles, rather, excitations of various quantum fields.
So what we consider a discrete photon particle with a location (x,y) would actually not be an individual particle and would be more like:
So what we consider a discrete photon particle with a location (x,y) would actually not be an individual particle and would be more like:
electromagnetic_field[y][x] = "excited"
Quantum fields are mathematical tools, they are not what the universe is made out of.
The measurable universe is not infinite. There's a limit to how far we can look back due to the speed of light, but the universe is expanding faster than the speed of light.
Aren't infinitudes accurate descriptors of any probability-based phenomenon? As far as I'm aware probability is not quantized.
There are supposed to be infinitely more real numbers than integers. But if you can name a real number, I can map it to an integer by converting the string you use to define the number to a bigint. This mapping suffers from mapping all definitions of pi to different integers, but that's OK. What is the mathematical term for "namable numbers"? That's what I'm talking about here, and it also seems related to the blog.
> What is the mathematical term for "namable numbers"
Well, it depends on what you mean by "nameable"—a common one is "Algebraic number", which is means "a root of a polynomial with integer coefficients". That's a bigger list that "those you can define using addition, subtraction, multiplication, division, exponents". But if you use a different list of operations, you get a different set of numbers.
Well, it depends on what you mean by "nameable"—a common one is "Algebraic number", which is means "a root of a polynomial with integer coefficients". That's a bigger list that "those you can define using addition, subtraction, multiplication, division, exponents". But if you use a different list of operations, you get a different set of numbers.
> When a mathematician says infinity they mean a repeatable process that we can keep doing forever.
I don't think many mathematicians would agree. The real numbers are an infinite set, but I don't think is commonly thought as something generated by a repeated process.
In common mathematics infinite sets exist just as much as finite ones.
In reality I can agree that infinite sets do not exist, but then neither do finite sets. At least, not in the way mathematicians mean.
I don't think many mathematicians would agree. The real numbers are an infinite set, but I don't think is commonly thought as something generated by a repeated process.
In common mathematics infinite sets exist just as much as finite ones.
In reality I can agree that infinite sets do not exist, but then neither do finite sets. At least, not in the way mathematicians mean.
What in the world did I just read? This should have stayed a drug-fueled thought, not a blog post.
The writer appears to be extremely confused, and tries to equate unrelated concepts.
The writer appears to be extremely confused, and tries to equate unrelated concepts.
By Löwenheim–Skolem, all models involving infinity might as well be countable, and if one limits expression size by the number of atoms in the universe, they might as well be finite too. Makes sense to me. :)
Here's one with tenure:
https://sites.math.rutgers.edu/~zeilberg/Opinion160.html
https://sites.math.rutgers.edu/~zeilberg/Opinion160.html
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Zeilberger's a lot of things, but 'confused' isn't one of them.
If there is a finite number of primes then there is a largest prime N>2. Construct N!+1. That is not divisible by 2..N, so it is prime.
Ergo, we say there are an infinite number of primes.
Mathematics is full of concepts that cannot be realized in the tangible world.
Infinite / infinity is one of them.
Spheres don't exist either - not in the mathematical sense.
Ergo, we say there are an infinite number of primes.
Mathematics is full of concepts that cannot be realized in the tangible world.
Infinite / infinity is one of them.
Spheres don't exist either - not in the mathematical sense.
Your proof isn't quite correct: N!+1 could be divisible by some number larger than N.
For example, if N=5, then N!+1=121, which is divisible by 11.
You can say that either N!+1 is prime or divisible by some prime larger than N. That means that there always exists some larger prime, but you haven't necessarily constructed it.
For example, if N=5, then N!+1=121, which is divisible by 11.
You can say that either N!+1 is prime or divisible by some prime larger than N. That means that there always exists some larger prime, but you haven't necessarily constructed it.
Yes, I messed up. Thank you.
> If there is a finite number of primes then there is a largest prime N>2. Construct N!+1. That is not divisible by 2..N, so it is prime.
5! + 1 = 121, which is not prime.
5! + 1 = 121, which is not prime.
Yes, I made a mistake. Thank you.
> a largest prime N
5 is not the largest prime.
5 is not the largest prime.
jwilk gave a counter-example which demonstrated my proof was incorrect (or at least incomplete).
You are mistaken. Your proof is correct. You assert that no prime less than or equal to N divides N!+1. The argument is clear. The other poster gives an example that does not gainsay that assertion.
From your assertion the contradiction follows that if N were to be the largest prime ...
From your assertion the contradiction follows that if N were to be the largest prime ...
I wrote "Construct N!+1. That is not divisible by 2..N, so it is prime".
That is not correct. 5!+1 = 121 which is not prime, as it is 11*11.
The correct statement would have been "so it is prime, or a composite with no prime factor <= N."
That is not correct. 5!+1 = 121 which is not prime, as it is 11*11.
The correct statement would have been "so it is prime, or a composite with no prime factor <= N."
You left out your "a largest prime." If N were indeed the largest prime then there would be no larger prime to divide into N!+1 and make it composite.
In any event it is a long way from wrong, IMHO. :-)
In any event it is a long way from wrong, IMHO. :-)
If my construction doesn't work for all arbitrary primes then it won't be guaranteed to work for "a largest prime", which means it doesn't provide an example of one way "infinite" is treated in modern mathematics.
(Remember, the original essay described a constructive requirement for "infinite" - “If you think there are infinitely many, write them all down.” I wanted to demonstrate how to think about infinity without that requirement.)
If someone in New Zealand sends me a letter and it's delivered to neighbor, it's still delivered to the wrong house - even if it gets to me eventually.
(Remember, the original essay described a constructive requirement for "infinite" - “If you think there are infinitely many, write them all down.” I wanted to demonstrate how to think about infinity without that requirement.)
If someone in New Zealand sends me a letter and it's delivered to neighbor, it's still delivered to the wrong house - even if it gets to me eventually.
Just wait until people find out that time doesn't actually exist either, other than as a measurement.
The MathOverflow commentator led off with "I’ve heard a worse story", meaning that not only do they not have direct knowledge of the situation, there's no guarantee that the person who told them the story had that knowledge either.
In other words, this is how urban legends spread.
In other words, this is how urban legends spread.
Here is a trick question, are there more natural numbers or more prime numbers?
Based on cardinality and infinity, they are both equal.
However, there are more real numbers than either.
Based on cardinality and infinity, they are both equal.
However, there are more real numbers than either.
But but but prime numbers are natural numbers and not all natural numbers are prime, ergo there is a cardinality flaw.
An infinite set can have the same cardinality as a proper subset of itself.
Two sets have the same cardinality if all their elements can be placed into a one-to-one correspondence with each other. For example, the positive integers can be placed into a one-to-one correspondence with the even positive integers: N <=> N*2
Two sets have the same cardinality if all their elements can be placed into a one-to-one correspondence with each other. For example, the positive integers can be placed into a one-to-one correspondence with the even positive integers: N <=> N*2
I remain unconvinced, given that we continue to explore the meaning and possibility of cardinalities beyond aleph 0. See, e.g., [1], and its implications for CH.
It strikes me that since we are still learning about cardinalities above aleph 0, we may have made mistakes at 0 itself and that there may be an “aleph -1”, an infinite cardinality less than aleph 0, which could apply, e.g., to the set of primes.
This would require proving that the bijection technique is not properly constructivist. That is considered constructivist has long bothered me, but I lack the detailed background in the field to articulate the objection.
[1] https://www.quantamagazine.org/how-many-numbers-exist-infini...
It strikes me that since we are still learning about cardinalities above aleph 0, we may have made mistakes at 0 itself and that there may be an “aleph -1”, an infinite cardinality less than aleph 0, which could apply, e.g., to the set of primes.
This would require proving that the bijection technique is not properly constructivist. That is considered constructivist has long bothered me, but I lack the detailed background in the field to articulate the objection.
[1] https://www.quantamagazine.org/how-many-numbers-exist-infini...
That's actually often used as the definition of an infinite set.
What is meant with 'something exists'?
"Vladimir Arnold [3] forcefully stated that it is wrong to think about finite difference equations as approximations of differential equations. It is the differential equation which approximates finite difference laws of physics; it is the result of taking an asymptotic limit at zero. Being an approximation, it is easier to solve and study. In support to his thesis, Arnold refers to a scene almost everyone has seen: old tires hanging on sea piers to protect boats from bumps. If you control a boat by measuring its speed and distance from the pier and select the acceleration of the boat as a continuous function of the speed and distance, you can come to the complete stop precisely at the wall of the pier, but only after infinite time: this is an immediate consequence of the uniqueness theorem for solutions of differential equations. To complete the task in sensible time, you have to allow your boat to gently bump into the pier. The asymptotic at zero is not always an ideal solution in the real world. But it is easier to analyze!"
From page 141 of Mathematics under the Microscope by Alexandre Borovik: http://eprints.maths.manchester.ac.uk/844/1/MicroMathWeb.pdf