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Infinity: Big and Bigger

Meet Georg Cantor, rogue mathematician and corrupter of youth.

by
Charlie Martin

Bio

October 17, 2013 - 4:00 pm

Georg_Cantor3

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, mind-bogglingly 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 3-1=2, and we know 2-1=1, and we know 1-1=0, but 0-1 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.

44_pythagorean_theorem

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 fast-forward 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 Russian-German 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.

shutterstock_149979035

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:

500px-Diagonal_argument_2.svg

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 n-th digit in our new number is always different from the n-th digit of the n-th 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.

shutterstock_41459863

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.

Charlie Martin writes on science, health, culture and technology for PJ Media. Follow his 13 week diet and exercise experiment on Facebook and at PJ Lifestyle

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Top Rated Comments   
The subject of infinity is very interesting and fun, with no end in sight.
1 year ago
1 year ago Link To Comment
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All Comments   (65)
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Well, my thoughts are that you were going along quite well, until you got to talking about different infinities. I was taught about the infinity of integers - that's countably infinite. I mean, you can count them, right? Assign a number to each integer as it comes along.....

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.
1 year ago
1 year ago Link To Comment
Except it would have been false. For every infinite set, there's a larger one.
51 weeks ago
51 weeks ago Link To Comment
The content of mathematics is zero; It says nothing about the real world.

Primitives do not count above three because they understand that there
are not that many things that are the same, not even days.
1 year ago
1 year ago Link To Comment
Which will surprise every engineer and scientist since Archimedes.
1 year ago
1 year ago Link To Comment
True, from mathematics alone, nothing can be proven about the world. However, a purely mathematical model - with variables whose particular values are assigned according to physical things that can be measured and with the help of physical experiment to establish its validity can say something about the world. The proof of that is physical - the existence of airplanes, television, nuclear reactors, high rises, computers, turbines and the like.
1 year ago
1 year ago Link To Comment
"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."

Sort of a metaphor for the regulatory state.
1 year ago
1 year ago Link To Comment
Consider that comment stolen
1 year ago
1 year ago Link To Comment
The point at which you get to integers and start to define negative numbers...I don't quite understand the logic around it. If you are describing the universe...well, the universe ceases at 0. That is the point when you have nothing.
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.
1 year ago
1 year ago Link To Comment
Well, there are a lot of things you can't have without zero. Like 10. And you use negative numbers all the time: credits and debits, positive balances and overdrafts, budgets and deficits, asserts and liabilities.
1 year ago
1 year ago Link To Comment
because we need numbers according to the task, for instance, Natural numbers are the numbers we (and the primitive humans) used to just count. It goes down to: 2 > 1 just because eating two apples leaves you with more food in your body, so you can associate a concept (a number) to something tangible.

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 potatoes-yet 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.
1 year ago
1 year ago Link To Comment
"...you have a proper scale described in Kelvin with no negatives." Well, maybe not. See following link about temperatues below absolute zero Kelvin.
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.
1 year ago
1 year ago Link To Comment
Remember that temperature is a measure of average speeds. You can't have a temperature below 0 kelvins because you can't go slower than standing still.
1 year ago
1 year ago Link To Comment
Yes, there are -ve temperatures on the Kelvin scale.

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.
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1 year ago
1 year ago Link To Comment
I meant to put in a link, but forgot.

http://en.wikipedia.org/wiki/Negative_temperature
1 year ago
1 year ago Link To Comment
If you had dark matter, that's not a negative...it's a different form of something. Electrons aren't negatives of protons.

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).
1 year ago
1 year ago Link To Comment
Zero is not an assumption, it is a definition. Mathematics is not built on assumptions, it is built on strict definitions and the application of logic. Zero is the number that represents the concept of nothing in mathematics just like the word "nothing" represents the concept of nothing in the English language.

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 anti-proton. 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.
1 year ago
1 year ago Link To Comment
you're totally right while the argument of negative Kelvin degrees just show how modern scientist have completely abandoned rationality:
* 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 counter-example, they proceeded to call it "paradox" instead of counter-example. This was the abdication of rationality. This is why you have aberrations like the "Drake equation" or the "String theory".
1 year ago
1 year ago Link To Comment
What is wrong with the Drake Equation? You are obviously trying to ascribe characteristics to it that it does not claim to have.

As for String Theory, it has proven to be a very useful model.

I bet you don't like Banach-Tarski balls.
1 year ago
1 year ago Link To Comment
Science is to Philosophy what a repair man is to an engineer, so according to your philosophic premises you are prone to accept certain ideas about the Universe you live in-and about how to deal with it.

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 conjecture-based even if there is nothing wrong-mathematicaly-with the Drake equation.

Also the Ptolemaic model was useful. It was mathematically beautiful as well and gave decent predictions, yet it was wrong.
1 year ago
1 year ago Link To Comment
The inclusion or exclusion of your silly parameter would not have any significant impact on N. But that is not the point. The Drakes Equation is a statement that expresses the type of the questions that need to be answered in order to get an idea of the chance of contacting an extraterrestrial civilization. It is not meant to calculate anything. The futility in being able to answer those questions is kind of the point. Its a thought exorcize.

You would be an idiot to use Relativistic Mechanics instead of Newtonian in order to put a sat in orbit.
1 year ago
1 year ago Link To Comment
As granddaddy used to tell me, if the bird book and the bird disagree, believe the bird. Your horribly irrational quantum theory, including the collapse of the state space on observation that Schödinger described, has stood up to every experimental test.
1 year ago
1 year ago Link To Comment
Except that pesky cat. Remember, I'm not criticizing the experimental tests (which are usually Classic physics, actuallly) but the interpretation given to it. BTW, there are like 20 interpretations, of which, the Copenhagen interpretation is only the most popular.
1 year ago
1 year ago Link To Comment
Nope, sorry. Entanglement and superposition have been demonstrated experimentally. Choose your interpretation, but the physical universe really does seem to have a property that means some events aren't determined until they're observed.
1 year ago
1 year ago Link To Comment
It's more accurate to say that the quantum state of an entangled system isn't determined until an observation is made. It's important to understand that nature itself cannot determine the quantum state of an entangled system until the observation is made. This isn't because the observation makes it real, it's because the entanglement leaves the system in a superposition of states and the observation forces the system into only one state. What state that is cannot in principle be known prior to the observation, but once a state is chosen, the rest of the entangled system instantaneously go into their correlated states regardless of how far apart they are.
1 year ago
1 year ago Link To Comment
Oh really? let's hear about that experiments! You're falling in that trap of "if you believe goblins make it rain, every time it rains is a proof that goblins exist". Thus, your experimental observations are not determinant of the validity of your theory. So, what's your proof of superposition? The double-slit experiment? You don't need Quantum mechanics to observe it, it is a classic experiment that is only "consistent" -as in "it doesn't refute our daydreaming". That however is way too far of being a proof.

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.
1 year ago
1 year ago Link To Comment
Your analogy with fish in a pond has nothing to do with entanglement or superposition. Entanglement is when the quantum states of a set of quantum systems cannot be individually determined because they all relate to each other and are dependent on each other. It's as if the individual quantum systems lose their individual identities to an overall state where all of them are hopelessly combined and it isn't mathematically possible to separate the individual states from the overall state until one of the individual states is determined by making an observation. You can't just make any observation, the observation must be specific to the entanglement. For example, if the entanglement is in the polarization of a set of entangled photons, then the observation must be a measurement of the polarization of one or more of the photons. As long as the photons remain entangled, nature itself can't determine their individual polarizations. The polarizations only become determined when an observation of the polarization is made on 1 or more of the photons. Instantaneously, and regardless of distance apart, the other photons are polarized according the correlations encoded in the entangled state.

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.
1 year ago
1 year ago Link To Comment
Finally somethig reasonable to read.
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.
1 year ago
1 year ago Link To Comment
"It is a very thin line between physics and methaphysics "

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 non-deterministic universe.

To pretend otherwise is not rational.
1 year ago
1 year ago Link To Comment
So now is you that start defending falsifiability in science? good to know.

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!!
1 year ago
1 year ago Link To Comment
"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."

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.
1 year ago
1 year ago Link To Comment
Oh for Gods' sakes look it up.
1 year ago
1 year ago Link To Comment
Lol. That's like Ptolemy saying: HEY! look! There are so many experimental instances where my theory is right. You can have all the experience yet fail to integrate and abstract physical laws from there. Again: your experiments are not the problem; your QM experiments are NOT a proof of QM, no matter how much scientists are repeating that since forever. The problem IS philosophical, and so is the interpretation given to the experiments.

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.
1 year ago
1 year ago Link To Comment
No, go look up Bell's Inequality. One of those papers shows that because of Bell's inequality superposition and entanglement are the only consistent explanation of the experimental evidence.

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."
1 year ago
1 year ago Link To Comment
Technically, Bell's Inequality can only rule out local realistic hidden variable theories. It can't rule out non-local hidden variable theories.
1 year ago
1 year ago Link To Comment
Information is energy. Observation is the transmission of information. The information is from that which is being observed.

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
1 year ago
1 year ago Link To Comment
You are not getting my point, so I'll try again:

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.
1 year ago
1 year ago Link To Comment
So you're establishing something as an axiom that doesn't correspond with the observed world. That's okat with me, but you should take it to the theology department; that's not science.
51 weeks ago
51 weeks ago Link To Comment
I get that you really don't understand the subject being discussed.

"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.
1 year ago
1 year ago Link To Comment
The mathematicians of a few hundred years ago would have agreed with you. We have a more abstract notion of number now. You can't argue from your physical notion of integer that modern usage is incorrect. Begin with a modern definition of number and you will find no contradiction. It's our right to define number so that negative numbers can be used to describe the universe, even if you do not like it. Far from being an odd way to describe the universe, modern science would be virtually impossible without negative numbers.

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.
1 year ago
1 year ago Link To Comment
Welll, you certainly seem to be harping on temperature...but I was talking about the scale. That the definition of the scale can make certain things happen within that scale.

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.
1 year ago
1 year ago Link To Comment
IF you think there exists parallel Universes, then sure I disagree for the philosophical reasons I wrote to you in another comment.

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.
1 year ago
1 year ago Link To Comment
Could someone please translate the above comment for me. He's gone over my head into the chaos region. I know I never said anything about 'the basis of modern mathematics relying on negatives' but I note that he *has* cleverly disproved everything I said about temperature by informing me that I was 'harping'.

ForTheWest signing out.
1 year ago
1 year ago Link To Comment
And I said I wasnt talking about temperature...I was talking about the scale. You can see that clearly by this sentence "Welll, you certainly seem to be harping on temperature...but I was talking about the scale."
1 year ago
1 year ago Link To Comment
I think the problem here is that you're trying to be a smart ass.
1 year ago
1 year ago Link To Comment
Nope. If I was trying to be a smart ass I'd be saying things like we need negative numbers to describe your IQ, but I'd never say something like that.
1 year ago
1 year ago Link To Comment
What's your deal? You don't like other people commenting or something? Did I violate your precious discussion bubble of talking about things on your terms?

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)
1 year ago
1 year ago Link To Comment
That's quite witty.
1 year ago
1 year ago Link To Comment
That being said, set theory isn't disproven. I think it is right...things do get bigger and bigger. You can kind of imagine it through the theory of parallel dimensions (except they aren't integers, but continuously connected). If you have a dimension that contains height, width, and depth. Well it gets infinitely big in 3 directions. You add time and it gets infinitely big in 4 directions, compounding on the other dimensions. But what if you have a dimension with life...such as we are in. Well, you could have a dimension where you move a pen an inch to the right...vs one where you don't. But wouldn't it also be a different dimension if you THOUGHT about moving the pen to the right vs not thinking about it?

It's infinite and continually infinite as more factors are added.
1 year ago
1 year ago Link To Comment
Well, you're heading in a different direction. Remember that Cantor's definition of two sets (or more) being of the same cardinality -- being the "same size" -- is that you can construct a one to one relationship between each element of the two sets. Cantor's diagonalization argument basically says "if you set up a list of *all* the natural numbers in the right way, mapping them to real numbers, I can show that I can always construct a new real number not in that list. Thus we can't construct that one-one relationship, and the two sets can't be the same size."

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 one-one 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
1 year ago
1 year ago Link To Comment
Sure it could be a different dimension - however, not to be insulting, but your thinking seems to be naive. You could learn more about what people actually mean when they use words for which you use uneducated definitions. This would mean studying mathematics and physics (and other modern mathematical disciplines). In the meantime, you are obviously thinking. Maybe there's a gem in there just waiting until you can express it in the modern idiom.

(BTW, in my comment above 'than' was supposed to be 'then'. My proofreader has been down with a bad cold).
1 year ago
1 year ago Link To Comment
Don't worry, I won't be a grammar Nazi. You're safe.

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.
1 year ago
1 year ago Link To Comment
42.
1 year ago
1 year ago Link To Comment
It is now understood that that was a mistake. It's 44.
1 year ago
1 year ago Link To Comment
For another good summary of this sort of things read "Gödel, Escher, Bach: An Eternal Golden Braid" by Douglas R. Hofstadter.
1 year ago
1 year ago Link To Comment
Yeah, I got a copy of GEB as a gift when it first came out. It made me what I am today.

I should sue.
1 year ago
1 year ago Link To Comment
I did also. It did terrible things to my 18 year old brain.
1 year ago
1 year ago Link To Comment
I read "The Mystery of the Aleph" many years ago. I admit it's not the definitive book on Cantor, but it blew my mind. For one thing, despite sucking at math, I more or less understood what Cantor was talking about. For another, I got my first glimpse of the vast "mental universe" that mathematicians inhabit. With Cantor, numbers felt like space. I felt a bit sad that I didn't have the mental equipment to explore it.
1 year ago
1 year ago Link To Comment
Charlie, you revealed the complete "incompleteness" of your conception of infinity under the rubrics of "big and bigger", presuming an added "ad infinitum" (which really means "towards" and not "arriving AT", which would be the case with "in infinitum"). You have concocted "potential" infinity, i.e., a process of serializing which contains no principle for termination, i.e., the process is extrapolatorily going on and on less an end, viz., is endLESS. All examples you offer (and Cantor & Co too) are extrapolatory, although Cantor & Co try to close the series off by treating potential infinity as a whole or totality (i.e., as terminally complete). The result is, as Laserowitz noted, a wonderfully interesting play with "as if" nuimbers or, in my terms, a manipulation of symbols whereby the foundation constitutes a contradiction (and it is really a humorous passtime to persue mathematicians who try to squate the circle, viz., totalize the endless), although the usage itself is consistent once symbol manipulation has begun.

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 Sino-Christian Studies (Taiwan) or at least send you the bibilographic info. I repeat my suspicion is that a Buddha-ist would find my analysis interesting, particularly its application to theology.
1 year ago
1 year ago Link To Comment
Cantor's diagonal proof is often described as a constructive process, but it is easily and more correctly described as an existence proof.
1 year ago
1 year ago Link To Comment
Yeah, hm, I wonder what Erret Bishop made of the diagonal argument. I vaguely recall he was agin it.
1 year ago
1 year ago Link To Comment
Okay, so *now* after that last comment the "show more" button works.

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.
1 year ago
1 year ago Link To Comment
True. Like the man said:

"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.
1 year ago
1 year ago Link To Comment
Wow, an Intuitionist, observed in the wild. I can't read your whole comment right now because the "show more" link is being obstinate, but my first question would be to ask whether you can state your exclusionary notion of infinity in mathematical terms? I think you're heading for Russell's Spanish Barber, the set of all sets that don't include themselves.

Like a lot of logicians, I tend toward the formalist position that these are first of all string-manipulation 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 symbol-manipulation 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.
1 year ago
1 year ago Link To Comment
The subject of infinity is very interesting and fun, with no end in sight.
1 year ago
1 year ago Link To Comment
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