Do Infinite Pencils Exist?
mitchgordon.me
mitchgordon.me
"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
Anyways, their reasoning in the rest of the article only holds if you assume the universe is finitely bounded in time and space.
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.
Modern mathematics is happy with actual infinities.
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...)
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.
https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Math...
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 are roughly 3,000 sexdecillion paperclips in the universe before you run out of universe and consume your paperclip-making systems into paperclips.
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.
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.
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.)
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.
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.
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.
Well, for all we know, they universe might be infinite and contain an infinite number of galaxies, stars, atoms, ...
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.
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"
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.
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.
The writer appears to be extremely confused, and tries to equate unrelated concepts.
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.
5! + 1 = 121, which is not prime.
5 is not the largest prime.
From your assertion the contradiction follows that if N were to be the largest 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."
In any event it is a long way from wrong, IMHO. :-)
(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.
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.
In other words, this is how urban legends spread.
Based on cardinality and infinity, they are both equal.
However, there are more real numbers than either.
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
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...