Infinity doesn’t exist. Some people say there is a difference between a “potential infinity” and “actual infinity”, but that seems like a wrong way to use language; it would be more precise to say that we can imagine processes that continue indefinitely—ignoring the technical details of running out of resources, heat death of the universe, etc. -- but nothing can be infinite here and now. Or, to put it shortly, “infinite” only exists as an adjective describing processes, but never as a noun.
There are also mathematicians that pretend otherwise and talk about various infinities, effectively extending their maps beyond the territory. But magic always comes with a price, and in this case the price is that there are multiplepossible maps (which all fit the territory, and only disagree about the things happening outside of it) and then the mathematicians start arguing about which one should they use, because there is no way to choose the “true” one (by abandoning the territory, all we can talk about is self-consistence, and you can have multiple different self-consistent maps), so instead they try to choose the most “elegant” one, which is kinda subjective, because even when someone finds weird things happening on a map, the owner of the map will proudly proclaim that this is not a bug, but a feature. -- This leads to a lot of confusion even among people who study mathematics, probably because the few people who understand what this is all about find it too obvious to mention.
Uhm, back to the point: infinity doesn’t exist. If we want to debate something meaningful, we can only debate properties of some indefinite process. For example, a process of “adding more small spheres inside the cube, while a ray shines at it diagonally”. But it is a process without an end. There will never be a moment when there are literally infinitely many spheres inside. We can only ask whether the process has some properties that converge towards some values.
Now I can answer your questions:
Really? What is the diameter of the sphere, which reflected the ray back? And how come, that a smaller sphere didn’t shadow this ray?
In each step of the process of “adding more small spheres inside the cube” it was a different sphere that reflected the ray. In each subsequent step, a smaller one. In each step the first sphere in the way of the ray wasn’t shadowed by yet smaller sphere, because the smaller sphere wasn’t there yet.
This is all there is. The cube will never be actually completed. Debates about what would happen if we send there a ray after it is completed… don’t necessarily have to correspond to anything real. By making the assumption that the infinity already happened we have left the realm of reality. Now it’s all about “my map is prettier than your map”.
But tell this to the 99% of the top mathematicians! For them all those infinities—infinite multitude of them—exist.
...I wonder whether this is true. I mean, as long as someone knows that you can have multiple mutually incompatible systems of axioms, then they should on some level realize that the choice of a specific system is arbitrary.
I mean, I haven’t realized this for a long time simply because I read have some popular books about infinities probably already at elementary school age, and they didn’t start with a big red warning “everything you will read in this book is merely a consequence of an arbitrarily selected set of axioms; other sets of axioms could lead to completely different outcomes; and the choice of the axioms is arbitrary because nothing of this is real, mwa-ha-ha-ha!” So, I naively assumed it was just business as usual. If I didn’t suspect books about prime numbers or complex numbers, why should I suspect the books about infinities?
And when I later heard something about the axioms, the idea of the “one true system of aleph-zero, aleph-one, etc.” was already firmly stuck in my head. So even while I was reading about how different axioms are possible, I still on some level assumed that the ones I was already familiar with are—in some sense—the correct ones (without asking myself what exactly the word “correct” could even possibly mean in this context).
Only a few years later I have finally connected the dots in my head. It was while thinking about natural numbers, the fifth Peano’s axiom, how without it we could have different models of natural numbers, and what exactly that means. And a few years later, my friend complained about the axiom of choice, and suddenly something clicked in my head, and I was like “wait, isn’t that the same thing as with the fifth Peano’s axiom? like you can have different models that all satisfy the same set of axioms, but they still differ at some underspecified places, and you need an extra axiom to distinguish between them?” Because at that moment I already had an understanding of what “having different models” means; it just never occured to me to apply it to infinities. (Despite seeing some hints which in hindsight obviously tried telling me to do so.)
In my defense, though, I don’t remember studying about infinities at university. Probably not even at high school, or at least not so deeply. So all my knowledge of infinities came from the popular books I have read at childhood; it shouldn’t be surprising it was seriously incomplete. I assume that people who actually studied this stuff at university, who were explicitly told about the existence of various axiomatic systems and given specific examples thereof, and then had exercises and exams… I believe they were in much better position to make the connection. So it would surprise me a lot if the best of them just couldn’t connect the dots even better than I did.
On the other hand, sometimes even highly intelligent and educated people fail to connect the dots.
I admit I am somewhat tempted to write an e-mail to Terence Tao and ask him what he thinks about this. There was never a better excuse to bother people much smarter than me. :D
I think you underestimate the zel of the 1000 top mathematicians about this infinity business. Perhaps there is about 10 dissidents among them, who are finitists.
But for all the others, the concept of infinity is a fundamental one. It’s easier to find a Catholic bishop who doesn’t care about God or Trinity, than a top mathematician who doesn’t believe and love infinity.
The infinity mythology is quite a rich one. Surprisingly rich, indeed.
First, we have the countable infinity—like all the natural numbers, or all the rational numbers. Aleph 0 is the cardinal of those. Then, we have 2^aleph0 or aleph1. The number of all real numbers. Between those two, you are free to postulate some greater then aleph0 and smaller than aleph1. You are also free to postulate that there is no cardinal between aleph zero and aleph one. It’s the Cohen’s solution of the so called Continuum problem.
You have more alephs above aleph 1. One aleph for every natural number. And you are free to postulate (or not) cardinal numbers between each two.
Then you have trans aleph big cardinal numbers.
Then you have non-constructible big infinite cardinals.
And then you have a lot of ordinals.
And then you have the Axiom of Choice. You can accept it or not. But it is proven, that if the whole structure of infinity is broken. then is broken with or without the Axiom of Choice.
I suspect Terence Tao has no strong opinion about ZF or ZFC system. He loves prime numbers and all the natural and real numbers and so on. But I don’t think he is in Set Theory very much.
Okay, I’ll try to be more explicit about this:
Infinity doesn’t exist. Some people say there is a difference between a “potential infinity” and “actual infinity”, but that seems like a wrong way to use language; it would be more precise to say that we can imagine processes that continue indefinitely—ignoring the technical details of running out of resources, heat death of the universe, etc. -- but nothing can be infinite here and now. Or, to put it shortly, “infinite” only exists as an adjective describing processes, but never as a noun.
(Historical trivia: I found this argument in Alfred Korzybski’s Science and Sanity. That’s the “the map is not the territory” guy.)
There are also mathematicians that pretend otherwise and talk about various infinities, effectively extending their maps beyond the territory. But magic always comes with a price, and in this case the price is that there are multiple possible maps (which all fit the territory, and only disagree about the things happening outside of it) and then the mathematicians start arguing about which one should they use, because there is no way to choose the “true” one (by abandoning the territory, all we can talk about is self-consistence, and you can have multiple different self-consistent maps), so instead they try to choose the most “elegant” one, which is kinda subjective, because even when someone finds weird things happening on a map, the owner of the map will proudly proclaim that this is not a bug, but a feature. -- This leads to a lot of confusion even among people who study mathematics, probably because the few people who understand what this is all about find it too obvious to mention.
Uhm, back to the point: infinity doesn’t exist. If we want to debate something meaningful, we can only debate properties of some indefinite process. For example, a process of “adding more small spheres inside the cube, while a ray shines at it diagonally”. But it is a process without an end. There will never be a moment when there are literally infinitely many spheres inside. We can only ask whether the process has some properties that converge towards some values.
Now I can answer your questions:
In each step of the process of “adding more small spheres inside the cube” it was a different sphere that reflected the ray. In each subsequent step, a smaller one. In each step the first sphere in the way of the ray wasn’t shadowed by yet smaller sphere, because the smaller sphere wasn’t there yet.
This is all there is. The cube will never be actually completed. Debates about what would happen if we send there a ray after it is completed… don’t necessarily have to correspond to anything real. By making the assumption that the infinity already happened we have left the realm of reality. Now it’s all about “my map is prettier than your map”.
I agree, that this is the solution. I completely and absolutely agree with you. It’s a false (mad if you wish) construction.
But tell this to the 99% of the top mathematicians! For them all those infinities—infinite multitude of them—exist.
What I don’t agree with you, is:
It cannot continue “indefinitely”—it will stop rather soon.
But yes … I gave this solution as the possibility number 6.
6. This is an illegal or ill defined question
You have answered correctly at the bottom line, Viliam. By my judgement. But that’s just my judgement, of course.
I disagree with your reasoning above the bottom line, but that’s not that important as your (and mine) final answer.
Okay, I am happy we came to an agreement, but...
...I wonder whether this is true. I mean, as long as someone knows that you can have multiple mutually incompatible systems of axioms, then they should on some level realize that the choice of a specific system is arbitrary.
I mean, I haven’t realized this for a long time simply because I read have some popular books about infinities probably already at elementary school age, and they didn’t start with a big red warning “everything you will read in this book is merely a consequence of an arbitrarily selected set of axioms; other sets of axioms could lead to completely different outcomes; and the choice of the axioms is arbitrary because nothing of this is real, mwa-ha-ha-ha!” So, I naively assumed it was just business as usual. If I didn’t suspect books about prime numbers or complex numbers, why should I suspect the books about infinities?
And when I later heard something about the axioms, the idea of the “one true system of aleph-zero, aleph-one, etc.” was already firmly stuck in my head. So even while I was reading about how different axioms are possible, I still on some level assumed that the ones I was already familiar with are—in some sense—the correct ones (without asking myself what exactly the word “correct” could even possibly mean in this context).
Only a few years later I have finally connected the dots in my head. It was while thinking about natural numbers, the fifth Peano’s axiom, how without it we could have different models of natural numbers, and what exactly that means. And a few years later, my friend complained about the axiom of choice, and suddenly something clicked in my head, and I was like “wait, isn’t that the same thing as with the fifth Peano’s axiom? like you can have different models that all satisfy the same set of axioms, but they still differ at some underspecified places, and you need an extra axiom to distinguish between them?” Because at that moment I already had an understanding of what “having different models” means; it just never occured to me to apply it to infinities. (Despite seeing some hints which in hindsight obviously tried telling me to do so.)
In my defense, though, I don’t remember studying about infinities at university. Probably not even at high school, or at least not so deeply. So all my knowledge of infinities came from the popular books I have read at childhood; it shouldn’t be surprising it was seriously incomplete. I assume that people who actually studied this stuff at university, who were explicitly told about the existence of various axiomatic systems and given specific examples thereof, and then had exercises and exams… I believe they were in much better position to make the connection. So it would surprise me a lot if the best of them just couldn’t connect the dots even better than I did.
On the other hand, sometimes even highly intelligent and educated people fail to connect the dots.
I admit I am somewhat tempted to write an e-mail to Terence Tao and ask him what he thinks about this. There was never a better excuse to bother people much smarter than me. :D
I think you underestimate the zel of the 1000 top mathematicians about this infinity business. Perhaps there is about 10 dissidents among them, who are finitists.
But for all the others, the concept of infinity is a fundamental one. It’s easier to find a Catholic bishop who doesn’t care about God or Trinity, than a top mathematician who doesn’t believe and love infinity.
The infinity mythology is quite a rich one. Surprisingly rich, indeed.
First, we have the countable infinity—like all the natural numbers, or all the rational numbers. Aleph 0 is the cardinal of those. Then, we have 2^aleph0 or aleph1. The number of all real numbers. Between those two, you are free to postulate some greater then aleph0 and smaller than aleph1. You are also free to postulate that there is no cardinal between aleph zero and aleph one. It’s the Cohen’s solution of the so called Continuum problem.
You have more alephs above aleph 1. One aleph for every natural number. And you are free to postulate (or not) cardinal numbers between each two.
Then you have trans aleph big cardinal numbers.
Then you have non-constructible big infinite cardinals.
And then you have a lot of ordinals.
And then you have the Axiom of Choice. You can accept it or not. But it is proven, that if the whole structure of infinity is broken. then is broken with or without the Axiom of Choice.
I suspect Terence Tao has no strong opinion about ZF or ZFC system. He loves prime numbers and all the natural and real numbers and so on. But I don’t think he is in Set Theory very much.