The nature of infinities
I have been thinking about the nature of infinity lately. I have no experience with higher mathematics or theorems regarding infinity, so please forgive me if my ideas on this topic are extremely naive or superficial.
Cantor's theorem proves that there are infinities that are larger in size than other infinities (again, forgive me if I have stated this wrong). This reminds me of someone who is given a titanium box, and there is a mysterious object placed inside the box. The person being given the box doesn't know what this object is, but he is allowed to conduct experiments on the box. For instance, he can measure the temperature of the box, its weight, and so on, but he is never allowed to directly (or indirectly) view the contents of the box. What Cantor has proven makes me think that humans to infinity is analogous to the man and the object. We have been able to discern some property of infinities, but we cannot view them directly. This is where the idea I have been thinking about comes in. As humans, we can automatically distinguish the magnitude of 3 and 6, and it is a simple task to see that 3 < 6. However, what if there were creatures on another planet, who found difficulty with this. Their numbers may be extremely small, in the limit as we approach 0 say, and they may be comfortable working with such numbers. They may even have theorems for their numbers, many of them, like we have Fermat's Last Theorem for the integers, and quadratic reciprocity for primes, and so on. In fact, many of our theorems may even break down with the "numbers" of these creatures, because the numbers are so small, that they do not even behave like our numbers. However, when they come to view the integers, or 0.5, or even 0.0001, they cannot view it directly, so they have to do what we did with infinity. They may use the tools they have, and deduce that one of these numbers is bigger than another, and so on. What if infinities are just extremely huge numbers, that they cannot be called numbers anymore. We cannot call them numbers, because you cannot treat $\infty$ as a number since it leads to contradictions, but what if they are really really huge numbers, i.e. that we actually need to define a new set which consists of all really huge numbers.
Question: Could infinity be just what we label as one of those large numbers, because we cannot see them or manipulate them directly? Maybe there are many of these numbers, like we have 1,2,3,..., each one bigger than the other. They may have their own algebra too, which is why plugging one of these large numbers ($\infty$) into our algebra leads to contradictions. Could there be a whole new mathematics just using large numbers, that we have no access to?
Furthermore, is it "correct" to say that, since our mathematics is entirely based on axioms which we view as self-evident because of the universe (for instance, if we lived in a new universe where taking a part from a whole leaves you with 2 wholes instead of "less than the whole", or whatever, then euclidean geometry breaks down completely), there could in theory be another mathematics based on these extremely large numbers, which complies with the physical intuition that the creatures who use it experience in THEIR universe? (which in turn explains why we cannot treat infinities as numbers in our mathematics, but they can)
I didn't want to post this on philosophy xchange, because they may not have the mathematical insight many of you hear have. Please reply genuinely and without shunning me off, I am genuinely interested, and as I said I apologize if this question is very bad, naive, or just plain stupid. I am curious and want to know an answer this way or that. Thanks.
infinity philosophy nonstandard-analysis
$\begingroup$ You've buried your question deep, so that if someone wants to figure out if they can answer it, they have to read a huge amount of text. Always start your question with some idea about the kind of question(s) you are asking, or you will drive off a lot of people who might be able to help you but don't want to read far to find out. $\endgroup$ – Thomas Andrews Aug 5 '14 at 21:18
$\begingroup$ @MPW: I don't think that's right. They might not be familiar with 3 and 6, but if you put three people here and six people there, it's clear which side has more people. $\endgroup$ – Asaf Karagila♦ Aug 5 '14 at 21:21
$\begingroup$ @Gobabis: You do know that "INFINITY" can be interpreted in several ways, some of which agree with your statements and others which don't. Right? $\endgroup$ – Asaf Karagila♦ Aug 5 '14 at 21:38
$\begingroup$ @Gobabis: No, it's not quite fundamental. And I can't give you a good explanation in a comment. This has been discussed before on the site, more than once. There are different types of infinity, and some of them don't even support an arithmetical system like that. Others do, in which infinities always "swallow" finite sizes, in which case your claim is wrong; and there are ways to interpret your statement that it is true, but there are delicate points to discuss here. And there is much more than would fit into a comment box. $\endgroup$ – Asaf Karagila♦ Aug 5 '14 at 21:47
$\begingroup$ I can call anything I like a number without contradicting anything. Contradictions happen when you do something more specific, such as assume the extended real numbers (which includes $+\infty$ and $-\infty$ along with the ordinary real numbers) obey the identity $x + 1 \neq x$. $\endgroup$ – Hurkyl Aug 6 '14 at 8:38
You asked numerous questions and provided a lot of information. I will keep my answer short and mainly refer you to other texts which may interest you.
Firstly, the cardinal numbers (which measure the magnitude of all sets, finite and infinite) can be considered as numbers, it's just that the infinite ones behave counterintuitively to us, at least until we get used to them and then their behaviour is less counterintuitive (obviously). All it says is that infinity is not something intuitive for us humans, and that makes perfect sense since we don't really encounter anything infinite in the world around us.
Your claim that $3<6$ is intuitively clear depends on context. You had to learn it. Kids learn to order small magnitude numbers pretty early on, but they need to learn it. Just to see how unintuitive it is, can you tell me which number is bigger $1000^{1000}$ or $1500!$? Once the numbers get big it is not your intuition that can help you order them.
I don't quite understand the analogy with the box. We have pretty concrete sets (e.g., $\mathbb N$, $\mathbb R$) for which we can argue quite directly about their cardinals.
In general the development of mathematics is deeply intwined and influenced by our experience of the world (see Mac Lane's "Mathematics, Form and Function" and Lakatos' "Proofs and Refutations"). If another civilization exists that experiences the world inherently different to how we do, it is likely that at least some aspects of their mathematics will be quite alien to us. If infinite quantities are fundamental to them, perhaps they will have developed cardinal arithmetic first and view our construction of the reals as some bizarre construction. Who knows.
I hope this helps. I strongly suggest reading both references above.
Ittay WeissIttay Weiss
$\begingroup$ Hmpf, typing on a crappy keyboard made me slower than you. :( $\endgroup$ – Asaf Karagila♦ Aug 5 '14 at 21:34
The OP's idea of infinite numbers as just extremely "huge" numbers that we have no means of accessing with our limited abilities resembles Leibniz's idea of an "inassignable" number. This is actually one of the motivating intuitions behind Edward Nelson's approach called Internal Set Theory.
Recall that Nelson is working in the usual ZFC, and in particular in the usual real line, but he is able to detect "huge" or "inassignable" numbers in the usual real line by means of a syntactic enrichment introduced into the language of ZFC.
Namely, one introduces a single-place predicate "standard" (perhaps a better name would have been "assignable" to echo Leibniz), so that if an integer is not "standard" then it is huge in the OP's sense, namely infinite for all practical purposes.
Being elements of the usual real line, these huge numbers satisfy the usual rules of algebra, to answer the OP's question specifically.
Mikhail KatzMikhail Katz
$\begingroup$ I don't think this is what the OP meant. It's clear from their preamble that by "infinite numbers" the OP meant infinite cardinals, not just large real numbers. $\endgroup$ – Jack M Aug 7 '14 at 10:18
$\begingroup$ @user45220 probably meant neither cardinalities nor hyperreals as his question is formulated at an intuitive level preceding mathematical formalisation. Why don't we ask him? $\endgroup$ – Mikhail Katz Aug 7 '14 at 10:26
Infinity, in general, is not a physical experience. If I removed just one molecule of water from a glass of water, or one grain of sugar from a spoon full of sugar, would you know? could you tell the difference?
Infinity, however, does simplify mathematical calculations quite a bit. And after some time where "potential infinity" was the only infinity in mathematics Cantor noticed that you can give a definition which works out nicely and introduced, in fact, set theory.
We still can't experience infinity. So this is nothing like the titanium box. We can't quite test it, or check its temperature. Moreover, if you do not subscribe to a Platonist approach that there is an actual existing universe of mathematics, then there can be several reasons why you can't even begin to understand infinity as a physical concept.
Mathematics, if so, may be guided by one of dozens of philosophies and intuitive ideas, but it is based on hard definitions and inference rules. So while we don't experience infinity physically in a way that we can discern the infinite from the "very large finite", we have definitions and we can derive conclusions from them.
My personal understanding of infinity is not something I can put into words, I'm sorry. It's somewhat visual and completely ineffable and intangible. But when I developed this intuition it was no different from understanding other people: you interact with them, and you try to guess what the result will be, and slowly you understand the characteristics of your fellow creatures.
Similarly here, we have definitions, and we guess that something is provable, then we try to prove it and it sometimes works out. Other times we show that there are counterexamples, and sometimes we show that neither is necessarily the case. Then, slowly we generate some intuitive understanding of infinity. But it's not like the box, because mathematical proofs and physical experiments are ultimately very different from one another.
Asaf Karagila♦Asaf Karagila
$\begingroup$ Just one comment. While mathematical proofs are very different from physical experiments, the scientific process guiding mathematics and physics is very similar (if not identical). The mathematical axioms, just like theories in physics, are trying to capture something. If the theorems/predictions of the theory disagree with our intuition then we may adjust our intuition but if the disagreement is too much to handle we change the axioms. $\endgroup$ – Ittay Weiss Aug 5 '14 at 21:42
$\begingroup$ Of course. We are guided by inductive reasoning, but ultimately a mathematical proof is deductive reasoning. Which is the key difference, other sciences use inductive reasoning and test whether or not this reasoning manages to make reasonable predictions of outcomes, but it's not quite deductive reasoning. $\endgroup$ – Asaf Karagila♦ Aug 5 '14 at 21:44
$\begingroup$ A mathematical proof, just like a prediction from a physics theory, is deductive reasoning. But mathematics, just like physics, is not only about proving things. It's also about building theories, choosing axioms, judging how useful a theorem is, and, most importantly perhaps, rejecting a theory because its theorems/prediction are (while proven) not correct. $\endgroup$ – Ittay Weiss Aug 5 '14 at 22:15
$\begingroup$ I think that we agree in principle, but as happened before, we argue over semantics. This sort of thing is not meant for the internet. We can only settle this in one of two ways: bare knuckles boxing, or having a beer. :-) $\endgroup$ – Asaf Karagila♦ Aug 5 '14 at 22:17
$\begingroup$ Next time I'm in your neighborhood... the beer is on me ;) $\endgroup$ – Ittay Weiss Aug 6 '14 at 0:22
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