Infinity: Big and BiggerMeet Georg Cantor, rogue mathematician and corrupter of youth.

October 17, 2013  4:00 pm

On the Internet, you can never go wrong by quoting the The Hitchhiker’s Guide to the Galaxy.
Space is big. Really big. You just won’t believe how vastly, hugely, mindbogglingly big it is. I mean, you may think it’s a long way down the road to the chemist’s, but that’s just peanuts to space.
Now, it’s kind of a cheat, because i’m not going to talk about that kind of space, I’m going to talk about spaces in a mathematical sense. But I’m offering something in exchange, because I’m going to talk about spaces that are much bigger than mere physical space.
The point of this is really to talk about (echo effect) infinity. And beyond.
Mathematically, space is much simpler than the thing in which your coffee cup is located just out of reach and that keeps your cat from being exactly where you’re sitting, no matter how much he tries. In mathematics, a space is simply a set of some sort with some kind of additional structure. (A set is just some collection of things with no duplicates, like {1, 2, 3, 4, 5}. By convention, we put sets into braces like that example.)
So far, that’s not a space — we haven’t said anything further about it than there is a bag full of things. But — since I’ve chosen a set we conveniently already know a lot about — we know that the set is ordered because we agree that 5 is bigger than 4. And we have a space.
Okay, it’s a pretty boring space, but it’s a space.
There are some other rules we think we know, like addition — 1+2=3. But in our little space, we immediately run into trouble, because 3+4 equals what? Oh, 7, but 7 isn’t in the set. To take care of 3+4, we need to expand the set to be at least {1,2,3,4,5,6,7} and then we’re immediately going to have the problem of 4+5, or for that matter, 7+1.
Now, with nothing more than the idea of addition (we talked about ordering, but we can define an order in terms of addition) we’ve run into our first experience with infinity. There is a set N that we can define like this:
 0 is part of N
 For anything that is part of N, which we’ll call n, n+1 is also in N.
We call N the natural numbers.
Now, N is pretty big. After all, no matter what n we pick, there’s always something bigger. This is what we call infinite. And all is well, until we think about subtraction: we know 31=2, and we know 21=1, and we know 11=0, but 01 isn’t in our set. So we define a new set called the integers which has new elements 1, 2, 3, and so on. We can throw in multiplication now, and all is good, but when we put in division we’re in trouble again: 2÷3 and 1÷2 aren’t in there. So we define another set called the rational numbers, Q.
Now, we’ve pretty much defined all the numbers anyone had any use for until the Greeks and Egyptians screwed it all up by trying to measure fields and distances.
The Pythagoreans in Greece were a sort of a religious cult that believed the entire Universe was made of whole numbers and rational numbers. Of course, famously the Pythagoreans proved the Pythagorean Theorem — usually we say “Pythagoras proved” but actually one of the rules of the Pythagoreans was that everything they did had to have Pythagorus’ name on it because he was “head of the lab,” a tradition that continues in some laboratories to this very day — anyway, they proved the Pythagorean Theorem, which showed the length of the hypotenuse of a right triangle was the square root of the sum of the squares of the other two sides.
Because the Pythagoreans knew everything was described by whole numbers and their ratios, they then spent much effort trying to figure out what ratio that square root was — until a fellow named Hippasus proved that number was irrational, it couldn’t be represented exactly by any ratio.
For which discovery the Pythagoreans promptly drowned him. He made the fundamental mistake of explaining it to the other Pythagoreans while on a boat trip; they threw him overboard.
Barbie was right.
We’re going to fastforward a bit now — we’re now looking at something with enough parts to talk about anything we’d describe with the normal geometry of the Greeks, what we call Euclidian geometry. Which gives me an excuse to quote one of my favorite poems, an Italian sonnet by Edna St. Vincent Millay:
Euclid alone has looked on Beauty bare.
Let all who prate of Beauty hold their peace,
And lay them prone upon the earth and cease
To ponder on themselves, the while they stare
At nothing, intricately drawn nowhere
In shapes of shifting lineage; let geese
Gabble and hiss, but heroes seek release
From dusty bondage into luminous air.
O blinding hour, O holy, terrible day,
When first the shaft into his vision shone
Of light anatomized! Euclid alone
Has looked on Beauty bare. Fortunate they
Who, though once only and then but far away,
Have heard her massive sandal set on stone.
The set we have now, of all the rational and irrational numbers, is pretty much all that we need for anything real, and so it’s called the real numbers, R. Oh, there was a little flurry of excitement when people started asking about the square root of negative numbers, but that was pretty easily disposed of with complex numbers.
But around the turn of the last century, there was a big effort on to sort of fill in the holes in mathematics and make everything axiomatic — reduce everything in mathematics to the sort of formal proofs that would satisfy logicians.
One question that turned out to be hard to answer was this: was the infinite set of real numbers “bigger” than the infinite set of integers? Was the set of rational number bigger than the set of integers? And what the devil would it mean to talk about the size of an infinite set at all? I mean, it’s clearly infinite.
Then along came a RussianGerman guy, Georg Cantor. He started thinking about these collections of things, starting out by just asking what it meant to talk about the size of a set at all.
Finite sets were easy — just count them, and you get a number called the cardinality of the set. {1,2,3,4,5} has a cardinality of 5. No problem.
Obviously you can’t come up with a number for the size of an infinite set — the thing you see called “infinity”, ∞, doesn’t act like a number in important ways — but he did realize there was a way to tell if two sets were the same cardinality, the “same size.” This is called the pigeonhole argument.
Imagine you have something like mailboxes, and you have a bunch of junk mail. Start putting the junk mail into the slots, one at a time. If you run out before you’ve annoyed everyone in the building, the set of junk mail is smaller than the set of slots; if you have leftover junk mail, the set of junk mail is bigger; and if you come out exactly even, the two sets were the same size. And you’re Goldilocks.
Mathematically, coming out even that way is called establishing a one to one correspondence. And there’s where the trouble started.
Cantor came up with a kind of proof called a diagonalization proof. It works like this: We lay out a set that is the same size as the natural numbers as a matrix like shown in the figure here:
Each of these has an infinite binary representation. Cantor used m and w, but it could be 0 and 1 as we’re used to looking at binary numbers now. So, the first row there is 0, which has an infinite binary representation pf ’0000…’. 1 is ’1000…’, 2 is ’0100…’ and so on.
Now go down the diagonal, and construct a new number, call it E, where the first digit is not 0 but 1; the second digit is 1; and so on. All we do is make sure the the nth digit in our new number is always different from the nth digit of the nth row.
Our matrix of numbers, we know, has all the natural numbers in it, and all of those numbers have their representation in binary. But now we have this new number, E, which also has a binary representation, but which we know can’t be equal to any of the numbers in our list.
(Why? By definition. If it were equal to some number in the list, say 32,260,309, then it would have the same digit at place 32,260,309. But since we constructed it by getting to digit 32,260,309 and inserting something that did not match, it can’t be equal. Pick any number you like; E can’t equal that number.)
This argument — there’s a longer explanation that I like here, and another here — and the definitions Cantor came up with to go with it, form what is now called set theory.
Cantor’s Theorem had a somewhat larger implication, though. What it meant was that any set you could construct had a bigger set that could be constructed from it. So, we had the set of natural numbers, and we knew there was a bigger set than that because we showed how to construct it. But if we can construct that bigger set, then we can use the same tricks to construct an even bigger set from that, and a bigger set from that, and so on. So, since we can always construct the next bigger set, we know that this set is infinite as well. There are infinitely many infinite sets.
Set theory is now seen as one of the foundations, perhaps even the foundation of mathematics, and Cantor’s method of diagonalization was later used by Kurt Gödel in proving his famous Incompleteness theorem.
It was very controversial at the time though. Very. Leopold Kronecker, a famous professor of mathematics at the University of Berlin (where Cantor had gotten his doctorate), said Cantor was a “scientific charlatan,” a “renegade,” and a “corrupter of youth.” Henrí Poincaré, another great mathematician of the time, called set theory a “grave disease” infecting mathematics. Wittgenstein later said set theory was “laughable” and “wrong.”
This was so tough on Cantor that he started to suffer periods of depression; these resulted in his repeated hospitalization and increasing debilitation until he died.
Lucky that Cantor didn’t first explain it while he was on a boat.
Now that's not really so bad, but then consider what happens when you try to count all the real numbers, rational and irrational. Darn if there isn't an infinity of them between 0 and 1, another infinity of them between 1 and 2...
This explosion of infinities is called 'uncountably infinite'. Now one might suppose you could count all members of these infinities, too, but you've already used up infinity but counting only the integers. Clearly this is a different kind of infinity.
And that, sir, would have been a better ending.
Primitives do not count above three because they understand that there
are not that many things that are the same, not even days.
Sort of a metaphor for the regulatory state.
The use of negative numbers with relation to describing the universe seems a bit odd. It's use is all well and good when you describe things like accounting, where the negative defines money owed vs money possessed. But with relation to everything, there is no negative.
The infinite at the 0 end is described by an exponential curve that theoretically approaches zero forever.
The point of this distinction is to point out the use of false scales. Most familiar with regard to our temperature scales. We have negative temperatures in Farenheit and Celcius, but when the true scale is set at a true zero, you have a proper scale described in Kelvin with no negatives. But also subject to false scale errors are the very use of things such as integers where they do not exist. Many scales are continuous, such as temperature, or distance.
I would also argue that 0 is not part of the set N. It is a false assumption. It's essentially saying nothing = something.
Negative numbers are used, when we started doing transactions, like: I will give you 10 eggs in exchange for your 5 potatoes; but the other day I gave you some good meat and wool in exchange of 10 potatoesyet you never paid me your potatoes; so I will take your 5 potates and you are still in DEBT of 5 potatoes.
on a primitive way, you can just list your debts in a separate list without needing that Negative sign, but as the life became more complex the nagative sign prooved useful in handling the operations/transactions.
http://www.sciencedaily.com/releases/2013/01/130104143516.htm
"But with relation to everything, there is no negative." I suspect there is a negative somewhere down there in the heart of a black hole that is a part of everything. Or maybe there's a negative everything out there/in there in that dark matter.
That's the one thing about science  it keeps changing. That's a good thing because it means we're still learning.
In thermodynamics, temperature is defined as dq/ds where q is heat exchanged reversibly with the system and s is consequent change in entropy. In a typical system, where there is no limit of energy states in which to add energy, this works out to average kinetic energy. However, some systems have a finite number of available states. Consider, for example, a system of electrons in a magnetic field. There are only two energy states per electron  spin up and spin down. The maximum entropy is reached when the electrons are half in the high energy state and half in the low energy state. Increasing the number of high energy states at this point actually decreases the entropy as can be easily seen by thinking of the system in its highest possible energy state  all electrons in the high energy state. There is only one way for this to occur, whereas at half high, half low there are many, so as total energy goes from half spin up, half spin down to all spin up (arbitrarily chosen here as the high energy state), the entropy actually decreases.
By, the 2nd law of thermo we can deduce that we cannot at this point, by heating with a body of finite +ve temp, increase the population of spin up electrons. Infinitesimal change of *energy* in the population corresponds to 0 change in entropy (a maximum). This means that by the thermodynamic definition of temperature, dq/ds, the temperature is infinite, and it flips from + infinity to infinity with an infinitesimal change of energy. (it's like "the" point at infinity in projective geometry). Anyway, as you increase energy after crossing this "infinity boundary", entropy decreases with increasing energy (the slope is negative) and this corresponds to a negative temperature.
Such systems can actually be put into a negative temperature state using laser pumping, for example. Energy will flow spontaneously from such an "inverted system" into another system no matter how hot it is, so it makes sense to say that the negative temperature system is hotter.
By average kinetic energy, you get the wrong prediction about which way energy will flow spontaneously. A container at 1,000,000K will spontaneously take energy from an inverted population in a finite number of states, even if, by average kinetic energy, its temperature is only 1000K.
This will probably make those who want to use 16th century definitions very angry. Who cares  we've moved beyond those days and we were right. How do we know? Computers, lasers, radio, rockets  that's how.
(show less)
(show less)
http://en.wikipedia.org/wiki/Negative_temperature
And yes, the Kelvin scale could have temps below 0 Kelvin if the scale needs adjusted, but theoretically the scale should end when all heat is absent and molecules stop moving. It's a theoretical end point. The article you linked sounds like a calculated finding...which could simply be the result of manipulating a scale. (Not saying it's absolutely the case, just thinking about the issues).
As far as you feeling that the use of negative numbers in relation to describing the universe seems a bit odd, then perhaps you haven't thought about direction of travel. For example, the measure of distance in the universe may be relative to an arbitrary zero point, however, the direction of travel can't be described without positive and negative. Clearly two space ships moving in opposite directions cannot be described without one direction being the negative of the other. No distance scale you can create can describe opposite directions of travel with only a positive number. Set the zero of your distance scale anywhere you like, set it where both ships are travelling only on the positive side of your distance scale. Regardless of where you set your zero, one of the ships will approach smaller numbers on the scale as it travels and and the other will approach larger numbers. If you subtract the position of the ships when they were together from the position of the ships when they are apart, one will have a negative distance and the other will have a positive distance. This doesn't mean that one of the ships traveled a distance less than zero, it simply means that the ships were moving in opposite directions. If you chose to always do your subtractions so that the distances came out positive, then you lose all information associated with their direction of travel, forcing you to then state that they were travelling in opposite directions in order to restore the information that was lost. In reality, you haven't eliminated the negative number, you simply made the "" sign much more verbose. On the other hand, you could try to use the distance between the ships as a measure of their differing directions of travel, with the idea that increasing distance equals opposite travel directions but then you would be fooled by 2 ships travelling in the same direction at different velocities.
As far as electrons go, no, electrons aren't the negatives of protons, as protons are composed of quarks (more on that later) and electrons are fundamental and, so far, indivisible. However, electrons are the negative of positrons, which are merely electrons with a positive charge. In other words, a positron is the antimatter counterpart to an electron. Protons also have an antimatter counterpart called an antiproton. In fact, each particle type has an antimatter counterpart, even a neutron. Neutrons and protons are composed of quarks. Antiprotons and antineutrons are composed of antiquarks. Quarks and antiquarks have opposite electrical charges and there are many different types of quarks.
Specifically, a Neutron is composed of 1 up quark and 2 down quarks. If we call the charge on an electron "e", then the up quark has a charge of +2/3e and each down quark has a charge of 1/3e. An antineutron is composed of 1 up antiquark with a charge of 2/3e and 2 down antiquarks with charges of +1/3e each. Add them up and you get zero electrical charge for both the neutron and antineutron.
Protons are composed of 2 up quarks and 1 down quark, for a total charge of +e. An antiproton is composed of 2 up antiquarks and a down antiquark, for a total charge of e.
Thus, having negative numbers, positive numbers, and zero are necessary to our understanding and teaching of the laws of physics. Yes, we could describe the laws of physics and the universe without negative numbers, but then we would spend all of our time writing extremely verbose things to replace the "" signs that simplify understanding and communication.
* There was a time when finding a contradiction (with a previous truth) indicated you just need to fix your premises.
* Now, modern "scientists" think they have revolutioned science and discarded logic.
If you want a example: Quantic Mechanics and the Schroedinger's Cat. Schroedinger just showed them how absurd according to simple logic was that Copenhagen interpretation... and, What the other said? Instead of recognizing the error with such a formidable counterexample, they proceeded to call it "paradox" instead of counterexample. This was the abdication of rationality. This is why you have aberrations like the "Drake equation" or the "String theory".
As for String Theory, it has proven to be a very useful model.
I bet you don't like BanachTarski balls.
That said, there is nothing wrong *from Mathematics point of view* with the Drake equation... except that it is based on assumptions unconnected with reality. To use a Popperian term, is not falsifiable. Just to show you, I can modify the Drake's equation to this:
N = R*Fp*Ne*Fl*Fi*Fc*L*Y
where Y is the Ytzik's Parameter, indicating the percentage of the time that recognizable signals from advanced extraterrestrial civilizations are absorbed by a wormhole, making it impossible to detect by humans... Oh... this also means we should need a "Density of wormholes parameter" and a "percetange of traversable wormholes". ALL conjecturebased even if there is nothing wrongmathematicalywith the Drake equation.
Also the Ptolemaic model was useful. It was mathematically beautiful as well and gave decent predictions, yet it was wrong.
You would be an idiot to use Relativistic Mechanics instead of Newtonian in order to put a sat in orbit.
You can't reasonably proof this: "the physical universe really does seem to have a property that means some events aren't determined until they're observed".
You don't KNOW how may fishes are in the nearest pond to your house UNTIL you go and OBSERVE them. That by no means imply they are or aren't there. If you think Existence is dependent on Conscience, or if you believe Existence is superior and independent of Conscience is a philosophical choice, and that's what defines whether a mind is rational or mystic.
In the example of the photons, it's important to note that it's not a case of the photons having polarizations that we just don't know about until we measure one or more of them, that's not the case. The case is that the polarization isn't set until the moment the observation is made. Only then do all of the photons instantaneously take on their correlated values of polarization. It's as if each photon doesn't know what polarization to go into until the observation makes the choice clear.
There is a big difference between what you are trying to debate vs. what Quantum entanglement and superposition really are.
You're right in that I'm not trying to debate the perceptuals and experiments of QM, but some underlaying philosophy in it. I know well what you're saying here and also know that there is a lot of practical applications of the quantum model that seems to give validity to it. However, I oppose the interpretations given without any logic to some of their results (logic as in Aristotle's logic), or without any connection or reality:
*Your measurement affects the state of a system? Fine.
*So you cannot know the original state of the system, because once you measure it, it is affected, and if you don't measure, you just don't know. Fine.
*You can construct a probabilistic model, fine.
*A wave function that collapses. Fine.
Yet nothing of it necessarily implies that the system didn't have a original well defined state, just that you can't know it. However, once the premise (philosophical premise) is changed to indeterminism as intrinsec in nature, then nothing can stop you from going wild. Then you start hearing people saying that ALL the states are actually happening but in other alternate/parallel Universes. It is a very thin line between physics and methaphysics and that's what dislike about Copenhagen interpretation. I may be in the minority part of people not accepting that interpretation, but that's fine. The truth is not a matter of popularity and it may take like 2 or more centuries to settle this thing forever.
Physics done on methamphetamine is physics nonetheless and I wouldn't be surprised if quite a bit of it was done that way.
"However, once the premise (philosophical premise) is changed to indeterminism as intrinsec in nature, then nothing can stop you from going wild."
No. It just means that science can't "prove" that free will, consciousness, a theistic god, and the like are nonsense. It doesn't follow that science can prove these things either. It actually never could anyway without accepting the illogical premiss that from a finite number of experiments one can conclude that an inviolable law exists (see "How to Learn from the Turkey" in Chapter 4 of "The Black Swan" by Taleb). But with QM, it is not possible to pretend that, even accepting that there are no black swans ignored in the characteristic inductive leap of science, that science rules rules out an nondeterministic universe.
To pretend otherwise is not rational.
Induction is not enumeration, is integration of the essential and separation from the accidental. Thus, the famous Black swan argument (defended by too many people) is invalid.
"To pretend otherwise is not rational".
yeah, just because you said it!!
Hmm, I wonder why I don't follow your argument.
"yeah, just because you said it!!"
No  the converse. I said it because it's true.
http://physics.about.com/od/quantumphysics/f/QuantumEntanglement.htm
http://www.pbs.org/wgbh/nova/physics/spookyactiondistance.html
Example: Heisenberg. we can't know precise position and velocity at the same time. FINE.
Interpretation 1: Ok, they have one but we can't know.
Interpretation 2: Ok, we can't know, thus they don't have one and the entire Universe is non deterministic.
:
add interpretations here.
If you prove the validity of entanglement (which is different than just "observe some entanglement here and there"), then you would do just that: Prove the entanglement, NOT a proof of Quantum Mechanics as an entire and consistent theoretical body of knowledgement.
Since we seem to talk different languages, I conclude with this:
QM remains a beautiful theory, just like Ptolemy geocentric model, full of Mathematics and parameters, etc, but a theory nonetheless, not yet proven, and it can't be accepted or refuted without a philosophical choice.
What you're saying is "there MUST be a hidden variable". What the math and the experiments say is "there can't be a hidden variable."
Before I became a student of Information Theory and Measurement Science, I was a student of Freshwater Ecology. Suppose a pond does indeed contain one fish right now. Prove it. If per chance you are able to do so, that knowledge become obsolete immediately. All you know is that there was a fish in the pond.
By the way, know what distribution is used to derive the number of fish in a pond from observation? Poisson's :D
Fishes are not CREATED by the mere act of observation. Their existence is independent of the observer, i.e., existence is above conscience (philosophically speaking). That's one of the principles refused in QM.
"Fishes are not CREATED by the mere act of observation" is a silly statement that is completely beside the point. And that point is the KNOWLEDGE of the existence of a fish in the pond.
When the magnitude of a phenomena is small relative to the impact of the observation needed to make a measurement of that phenomena, a minimum indeterminacy limit exists that is greater the zero.
As far as temperature goes, temperature was defined in a way that an 'absolute' zero is implied (not 'true' zero, which has a mystical connotation about it of the type that science abhors). Temperature could just as well have been defined as the logarithm of what we call temperature, which in some ways might have made more sense, it being harder to get from room temperature to .001K than it is to get from room temperature to 10,000K. If the log of temperature as defined today was used than approaching 0K as currently defined would be to approach infinity in the log approach.
But ve temperatures K are achieved by physical systems anyway. See
http://en.wikipedia.org/wiki/Negative_temperature
Don't argue with this using your personal definition of temperature. Learn the modern definitions, and it will all make sense.
I'm not sure what you mean about the basis of modern mathematics relying on negatives. You reinforce what I basically said about defiining the scale. I'm talking about the context of the discussion. If you're talking about everything in the universe relating to philosophy, it does matter how you define things. If you are talking about what is possible within a mathematical system that you created. Sure, go wild and make anything happen.
Of course Mathematicians can go wild and anything could happens, you can even define "unconventional operations" so to make a Set become a Space with certain operations (of addition and multiplication) and certain properties. Yet, that is not always reflected in Nature. The final proof of an Existence is that you have to show the entity... like the Strings in String theory, or that belief in parallel universes. THIS is also when Godel got wrong. An Axiom of Choice is NOT a blank check and it is not without consequences.
ForTheWest signing out.
You're pretty much just coming off as a whiny little brat. Your IQ joke isn't even original.
Your slogan should be "Think inside the box"
(That box being a theoretical box with a single particle in an ideal system...or your mom's vagina)
It's infinite and continually infinite as more factors are added.
But if you have a space with dimensions that are real numbers, the cardinality of that set of real number vectors is still the same cardinality as the reals; you can always construct a oneone relation between them. And you can prove that if you can do that for _n_ dimensions, you can do it for _n_+1, so it's true no matter how many dimensions you have.
However, it *does* turn out to be useful to think about those spaces of (countably) infinite dimension. They're a kind of Hilbert space, and important to a lot of areas of mathematics. http://en.wikipedia.org/wiki/Hilbert_space
(BTW, in my comment above 'than' was supposed to be 'then'. My proofreader has been down with a bad cold).
How exactly is my thinking being naive? (Not to be insulted or anything).
What exactly do you consider uneducated definitions? (Again, not to be insulted or anything)
And I have studied mathematics and physics, but I'm taking this from a philosophical approach to apply some context to the mathematics.
I should sue.
The extrapolative method starts with one unit, number 1, and continues on and and on ad infinitum (= always or endlessly "bigger", but never arriving thereAT as is the case of "in infinitum"which is the amusing goalless goal). I suggest an "exclusionary" method (found in Nicolas of Cusa) for defining ACTUAL infinity, namely as that which EXCLUDES all finitude. In other words, treat actual infinity in the terms of excluding any finitude, including a beginning and end, i.e., my method excludes "endless" (or your "big and bigger") as being included in the actuality of infinity. (With all seriousness, I tentatively think Buddha would have liked my approach.) This approach leads to the opposite conclusion of Cantor & Co, namely there are infinities of infinity. The conclusion is that any use of finitude, including the number 1(,2, 3, ...) cannot be logically applied to actual infinity. Actually there is NO number (or if you will, 0 number) of actual infinities, but actual infinity IS! This thesis is a paradoxical, but not a contradictory conclusion.
Before you declare me a clown, you have my email address. Contact me and I will try to send you per attachment a copy of my article in the Journal of SinoChristian Studies (Taiwan) or at least send you the bibilographic info. I repeat my suspicion is that a Buddhaist would find my analysis interesting, particularly its application to theology.
Let me just say I don't consider you a clown at any time  I just think that, in the context of the Buddhism column, the philosopher game is unproductive, akusala. I guard against them over there because I like them *too* much.
Over here in science and math, now, we're playing a different game.
I can't easily access your email address, I need greater permissions than I have for the Lifestyle blog; but you can contact me at ask.charlie.martin@gmail.com. I would be interested in seeing the paper.
"This is called a thicket of views, a wilderness of views, a contortion of views, a writhing of views, a fetter of views. Bound by a fetter of views, the uninstructed ... is not freed, I tell you, from suffering & stress."
Knowing lots of stuff about Infinity didn't free G. Cantor from suffering.
Like a lot of logicians, I tend toward the formalist position that these are first of all stringmanipulation games; it turns out that some of them, like arithmetic, have an interpretation that turns out to be useful.
Maybe the deepest result of 20th century mathematics is that built up via Cantor, Gödel, Turing, and Chaitin, that those symbolmanipulation games have inherent limits, that there are questions that can't be decided, things that can't be computed, models that can't be simplified, and then via complexity theory, "chaos theory", that there are apparently simple physical systems that *also* can't be exactly computed, systems for which no simpler model exists than the system itself.