Infinity in an Hour

- How it Feels to study Pure Mathematics

To see a world in a grain of sand,

And a heaven in a wild flower,

Hold infinity in the palm of your hand,

And eternity in an hour.

- William Blake

I have a master in pure mathematics (it also includes a substantial amount of applied mathematics and physics). People often wonder what is it that we do? And they visualize the most complicated mathematics they know, and they try to take it to the next level mentally. But it is very hard to try to understand something you know nothing of. People who only have a minimum of mathematical knowledge guess "Oh, so you multiply really large numbers", those who had math in high school/college guess "Oh, so you differentiate and integrate difficult functions?", those who have a university education where they needed advanced mathematics guess "Oh, so you solve hard differential equations on difficult grids/spaces?". Today I will endeavor to give a picture of how I feel when doing pure math.

For inspiration, I recommend the following videos. After we have seen them, we will try to do some of this mathematics. Note that we mention Cantor several times.

http://video.google.com/videoplay?docid=-8492625684649921614&hl=en

(Part 2 if you're interested)

http://video.google.com/videoplay?docid=-1663091361786740235

Pure mathematics is that of unlimited abstraction and precision. We define everything as precisely as humanly possible, and we try to make everything into abstract concepts. Understanding something in pure mathematics is often like walking a tightrope over an abyss, no room for small deviations or imperfect intuition.

A lot of the problems I think about when working, you would need 4 years of university education just to be able to ask. But now I will use some examples that you need almost no mathematics education to understand.

First we will look at sizes of infinity (this is very close to the paradoxes talked about in the videos above). To talk about size, we have to define it, and size of what?

Definition A set contains elements, and an element is contained in a set.

Example {1,2,3,5} is a set containing the elements 1, 2, 3, and 5.

Example 2 {a, b, car, boat, 99} is a set containing the elements a, b, car, boat and 99.

Definition 3 The size of a set is the number of elements it contains

Example 4 The size of {1,2,3,5} is 4

Example 5 The size of {a, b, car, boat, 99} is 5

Using these definitions, we could have defined a set to be infinite size if it has no finite size. But to be able to speak of different infinity sizes, we have to use a more fine tuned definition.

Definition 6 Two sets are equal if there is a bijection between them

Definition 7 A bijection is a function which maps all the elements of a set to the elements of another set, and where no two elements are mapped to the same.

To understand this last definition, you should see some examples. So look at wikipedias page bijection.

For more information, see function and Bijection-Injection-Surjection.

Now we can state the first question

Question 1 Are there as many even (positive) numbers as there are (positive) numbers in total?

Answer 1 (with proof) Yes. Let E={2, 4, 6, 8, 10, ...} be the set of even numbers, and let N={1, 2, 3, 4, 5, ...} be the set of all numbers. Then we can construct a function, f, from E to N by f(x)=x/2. This function is a bijection. Hence the size of E is the same as the size of N.

How can this be? Clearly the set N contains the set E, and then some, how can they be equal? Well we just proved that they were. The only thing we can conclude form the fact that N contains E is that the size of E is smaller or EQUAL to the size of N.

What we just did is known as Hilbert'sparadox of the Grand Hotel, with Infinitely many new guests. You should look at the link. This is NOT a paradox, this is a well established mathematical fact. What seems to be a paradox is only because our intuition is not used to the concept infinite.

Calculations with infinite

If you use a modern computer program, it often has inf (infinite) as a kind of number. It will normally give the following computations:

inf + 1000 = inf

inf – 1700 = inf

inf + inf = inf

inf*1000 = inf

inf*inf+inf^inf = inf

1/inf = 0

1/0 = inf

inf-inf = NAN

inf/inf = NAN

(-1)^inf = NAN

Here NAN means Not A Number. That is because you are not allowed to do these operations, the result is undefinable (if you chose a definition you would end up with a contradiction). The problem with inf-inf is that we don't know which inf is "largest". In some applications you may get the answer 0, in others 31, in yet others you may get inf.

Countable and Uncountable infinity

The size of the set Z (all finite numers), or the set N (all positive finite numbers), is called countable infinity (the sizes of Z and N are equal). It is easy to prove that the set of all fractions, the rational numbers Q, also has countable size (see http://www.homeschoolmath.net/teaching/rational-numbers-countable.php).

But what about the set R of real numbers (all numbers, including pi and the square root of 2, but not including imaginary numbers), is this also the same size as Z and N and Q? No. This proof is quite deep, and took me several days to understand (several years ago). If you want a challenge, see wikipedia's page on Cantors diagonal argument.

The Length of the Rationals (the set of fractions Q)

There is another very much used notion of size. This is what we use for integration, and to avoid confusing it with the size of sets from before, we call this new thing for length.

Definition The length of an interval on the real line, is the right endpoint minus the left endpoint.

Example We write [-3,7] for all the numbers between -3 and 7 including -3 and 7. The length of this interval is 7-(-3) = 10.

We can generalize this concept of length to other sets than intervals, for example to the union of intervals. Not surprisingly we get:

Theorem If one set is contained in an interval, the length of the set is smaller or equal to the length of the interval.

Supertheorem: The size of Q is countably infinite, but the length of Q is 0 (on the real line).

Proof: We have already seen that the size of Q is countably infinite. What about its length? Well, write Q as a sequence Q={q1, q2, q3, q4, ...} where all qi are fractions. Let K>0 be any arbitrary number (for example 0.000000001). Then the first fraction, q1, is contained in an interval of length K, namely [q1-K/2, q1+K/2]. The second fraction is contained in an interval of half that length (namely [q2-K/4, q2+K/4]). The third fraction q3 is contained in an interval with half that length again. Let us sum up this:

q1 contained in an interval of length K

q2 contained in an interval of length K/2

q3 contained in an interval of length K/4

q4 contained in an interval of length K/8

q5 contained in an interval of length K/16

...

So Q must be contained in the union, which will be an set with size smaller than (smaller because some of the intervals may overlap):

K + K/2 + K/4 + K/8 + K/16 + ... = 2K

How to calculate this? This is what we call a geometric series.

What do we now know? Q is contained in something of length 2K (you can choose any K>0). Hence the length of Q is smaller or equal to 2K (you can choose any K>0). The only possibility is that the length of Q is 0. QED.

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Comment by Nok er Nok:

I'll keep beating the same horse as always, I guess. We have spent some time discussing these questions and generally agree, but I can't help but object to the following paragraph. I have mostly the same knee-jerk reaction to it as the Newcomb's paradox, Hangman's paradox, etcetera. On the whole, I'll even illustrate my point with an XKCD strip (gasp!),  http://xkcd.com/169/   , though I'm not sure if the miscommunication is intentional or not.

"What we just did is known as Hilbert'sparadox of the Grand Hotel, with Infinitely many new guests. You should look at the link. This is NOT a paradox, this is a well established mathematical fact. What seems to be a paradox is only because our intuition is not used to the concept infinite."

This is most certainly a paradox. The mathematical formalism you describe and which mathematicians use is consistent and useful, yes, that's not what I'm trying to deny. Using that formalism to claim there are -as many- even numbers as natural numbers, however, is dishonest; using layman's terms in that manner -creates- the paradox. Nobody protests against the bijection-wizardry which is firmly belonging to alternate math-dimension. They do, however, have issues with mathematicians redifining common English words to mean something entirely different, and then marveling at how counterintuitive it is - particularly when they a few moments later set the trap by mixing real world example, impossible mathematical constructs, normal English and indistinguishable mathematical definitions.


Let's deconstruct Hilbert's Hotel to start with, and let's pretend we are not familiar with the mathematical formalism for dealing with infinities. Hilbert then claims the following:

 - hotel with infinite number of rooms
 - infinite number of guests at the hotel
 - such that all rooms are occupied, each by a single guest
which is all fine and dandy, but sets up for the counterintuitive conclusion
 - the hotel can still house another group of guests, exactly as large as the number which currently occupies every single room

Now, if you look at that without consulting your English<->Mathematics dictionary, you will surely conclude that this is perfect nonsense. Somebody is pulling a cheap parlour trick on you, one of these words have to mean something else that is appears. You can easily construct equally (more?) valid arguments than Hilbert presents, to show that the conclusion is impossible. For instance, it is easy to visualise that no room will be left unoccupied after you swap any two guests between their respective rooms, any number of times, and even though Hilbert does this an infinite number of times, this shouldn't change anything.


The parlour trick here is, of course, that occupied does not mean occupied at all, it has to do with bijections, and infinity does not mean a number you can increment arbitrarily many times, it is instead some mathematical construct with such properties that it cannot possibly have anything to do with any actual hotel. You might say the point of Hilbert's Hotel is that infinity cannot be treated as just any large number, you claim that our intuitions are not prepared to deal with infinity, but I strongly disagree. Hilbert's Hotel only shows that the mathematical lingo he ends up translating to 'occupied' and 'infinity' has nothing to do with a normal understanding of these words.

Taken as a story to accompany the mathematical formalism, to illustrate how it handles infinities, cardinality and size as something to do with bijections, it does an okay job. Without that context, it is not the slightest bit clever or enlightening, but just a load of gibberish. Hilbert's Hotel says -nothing- -whatsoever- about how hotels of arbitrary size work; it intentionally mixes mathematical formalism with a real world example which it then -fails to describe-!


Of course, all of this comes from the same sort of people who with a straight face will call f(x) = constant  an increasing function - and a decreasing one, at the same time. Nevermind that increasing is a code word for non-decreasing, which you cannot know without consulting your Google-translate English<->Mathspeak, or being familiar with the tradition of inclusive definitions in mathematics.

I think inclusive definitions are useful, and I'm not quite decided on whether using somewhat familiar but inaccurate and misleading terms is better than inventing new, arbitrary ones. However, I'm certainly not going to give mathematicians any credit for clever paradoxes which does nothing but illustrate that the mathematicians themselevs do not understand that their redefined words cannot be inserted into common English prose without appropriate and careful translation.


As an endnote, I feel fairly certain that it would be very possible to develop a formalism in which the natural numbers, the even numbers, the prime numbers and so on and so forth were -not- the same size. Of course, these alternate defintions would not develop fruitfully into integration and cardinality, like the current one does, but this alternate mathematics would be able to present the exact same Hilbert Hotel and the exact opposite conclusion; the hotel -cannot- accomodate even one more guest, much less another infinity of them. Or perhaps they would balk the moment you suggested that every one of the infinite number of rooms is currently occupied. And if this is true, it should be all the more obvious why Hilbert's Hotel is, indeed, a paradox of sorts.  

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Solving Conundrums 3

- Solutions to Solving Conundrums 1 (and 2)

The Second Conundrum

"Which way would you tell me to go if I were to ask?" (Then take that way)

If it's the one who always lie, he would tell you to go the wrong way, but as he is lying he has to lie to the question by telling you the right way.

The Fourth Conundrum

The solutions to this depends on what you mean by surprised. In practical terms, what the death-sentenced king does is to believe with all his mind that he will be killed tomorrow (he believes this every day). Tuesday morning he is not killed, but does he consider himself to be surprised?

There are three ways to define surprised (that I can think of). The first is "I am surprised whenever I am wrong". Using this interpretation, where is he wrong in his analysis? He must decide in advance on one day when he thinks he will be executed. If he decides 'I think I'll be executed on Tuesday', then he is surprised when he is executed on Friday (or, rather, when he is not executed on Tuesday).

The second interpretation is "I am surprised whenever I have a false negative, that is, when I predict that I will not be executed but I am." So if you have a false positive (you think you'll be executed but you're not) you don't count as surprised. Then the king is correct, his analysis is good, it is impossible to surprise him. (This goes against the way I told the story, but there exist different versions.)

The third interpretation is a bit more complex. If it rains/not rains tomorrow, are you surprised? Not too much, because you don't have a strong prediction. But if a volcano erupts in your neighborhood, then you are surprised. If someone asked you beforehand you'd say that the chance of a volcano erupting so near you was almost 0, or just say 0. The interpretation is: "You are surprised whenever something occurs that you assigned less than 10% probability" (I chose 10% for convenience, you can substitute it with any number less than 50%). Every night the king can assign 50% probability to being executed the next morning. Then he is never surprised.

The Third Conundrum

When making statements in logic you can make almost any statement you can dream up. Almost. You are not allowed to define a logical variable by 'P:(not P)'. The easiest way to make sure your statements are definable is to only use logical variables that you have already defined, together with logical operations (and, or, not, etc.).

Let us define the logical variables:

P: not Q

Q: not P

This is impossible in the same way as

P: not Q

is impossible (you can't know if P is true or false, it can't be true or false, as you have not even defined Q).

If you'd rather think of this as a satisfiability question (see the last solution on this post) for 'P equivalent (not P)', then the answer is "no, there is no truth values satisfying this expression".

The First Conundrum

Using the clarified version:

If A) is correct then D) is also correct, and then A) is not correct as there are two correct solutions.

If B) is correct, then B) is false by its own statement.

If C) is correct, then C) is false by its own statement.

If D) is correct then A) is also correct and then D) is not correct.

So every answer gives a contradiction, hence none of them are correct. But isn't then alternative C) correct, since no of the alternatives are correct?

The resolution here comes from thinking about satisfiability. If you have a logical system you can give a number of variables A, B, C, ..., and a logical 'equation' (statement), for example

(A and B) or (C and (not A) and (not D) or (A equivalent B),

and ask the question "Is there a set of values for the variables that make the equation true?". The answer can be

"yes; A=true, B=false, ...", or

"no; there is no such choice of values".

Finally, the answer becomes "The question you posed has no correct answer". On this higher level, where you defined the question, you can say that there is no answer that can be correct (as all of them leads to contradictions).

The question cannot be satisfied. It is not so that 0 of the alternatives are correct on the level of the question. On the level above, it is so.

If you disagree, I would recommend a book on mathematical logic, and one on set theory (not naive set theory, but the serious kind), at least read the wikipedia articles

http://en.wikipedia.org/wiki/Satisfiability

http://en.wikipedia.org/wiki/Propositional_satisfiability

And to see how complicated stuff becomes when someone tells you everything, see

http://en.wikipedia.org/wiki/Russels_paradox

http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory

Gambling All Of Mathematics

"A chess-master may gamble a piece, or even the entire game; but a mathematician writing an ad absurdum argument is gambling all of mathematics." - Couldn't find the source.

- This post can also be considered a hint to Solving Conundrums.

What is an ad absurdum argument? As always, you can see wikipedia, but let me take one of the most famous examples, and then give an explanation. Essentially the argument goes like this: Assume the opposite of what you want to show, arrive at a contradiction, conclude the opposite of what you assumed.

Example of an ad absurdum ("to the absurd") argument:

Are there a finite or infinite number of primes? (Primes are numbers p that cannot be factored into any other factors but p and 1. Primes per definition are > 1.)

Well, assume the statement Q to be true:

Q: 'There is a finite number of primes'

Then there are exactly n primes for some number n, and we can number these primes p_1, p_2, p_3 ... to p_n. Then we make a new number s by multiplying all these primes, and then adding 1 (s = p_1*p_2*...*p_n +1). This new number s will not be divisible by any of our primes, so the only possible factors of s are itself and 1. Hence there is at least n+1 primes. But this is impossible as n is the total number of primes.

We assumed Q and ended up with a contradiction, so Q must be wrong. Hence there is an infinite number of primes.

Example end.

So how does this work? I assume that we agree on what Logic is (the axioms and rules), as it would be far too cumbersome to write it out. Last time we talked about consistency of logic. This can be represented by the following axiom:

((P) and (not P)) equivalent to false

What we got in the proof was that the number of primes was exactly n and n+1, that is, exactly n and not n. What we got was equivalent to false by the above axiom. We showed that:

Q implies false

Now we want to use something called the rule of transposition:

(A implies B) is equivalent to ((not B) implies (not A))

In total we get

(not false) implies (not Q), which gives

true implies (not Q), which gives

not Q, which is

'There is an infinite number of primes'

“Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth.” - Sherlock Holmes

Have Mercy on my Implication

There is some sayings that irritates me. Like, "this is a quantum leap" (Norwegian: "Dette er et kvantesprang"). A quantum leap is the smallest possible leap in nature, in everyday life a quantum leap is the same as a continuous increase, as it is extremely small. That can be remedied, however, by thinking of the quantum leap as a leap in understanding. Physics at the scale of atoms were poorly understood until the concept of a quantum (the world is not continuous, but fundamentally quantized).

And then there are some that have no valid interpretation, some phrases that want me to knock someone's teeth out: "Yes, that implies truth."

It is common to confuse the following three words:

Correlation: Often when there is sun and rain we see a rainbow.

Causation: If you hit me on the head, I will feel pain.

Implication: If the moon is made of cheese, then I am eating chocolate right now.

(Since the moon is not made of cheese, the implication is true whether I am eating chocolate or not. Note also that moon and chocolate have nothing to do with each other.)

See Wikipedia's three pages for a more thorough explanation. Some may argue that implication is an abstract model for causation, but let us avoid philosophy right now.

Back to business: Instead of saying "X is true" some people say "X implies truth" to sound more wise. (In Norwegian: "X medfører riktighet"). This is completely bollocks! If you know that X implies truth, then you know nothing at all about X. ANYTHING implies truth! (Can you feel my frustration?) On the other hand, if someone were to say "Truth implies X", then you would know that X was true.

The Limitations of Logic

- Consistency, completeness, and Gödel's theorems.

- This is not really a hint to SolvingConundrums, but it is a prerequisite for understanding some of the solutions properly.

What is a logic? What is a mathematics? It is a set of axioms and some rules for deducing true statements.

Logic is the set of rules and axioms we have agreed to use. I will assume that we agree on these. (I refer to the mainstream choice, I know there are other candidates, like fuzzy logic, and I may come back to that later.)

A consistent system is one in which not both 'P' and 'not P' is true. In other words, a system where there is no statement P that is both true and false. Why is it so bad to have one such statement in our logic? Well, assume that we have one such statement, and call it P. Then for any other statement Z we get that P implies Z, because P is false. Then, since P is true, we can deduce that Z is true. The conclusion is that every statement in our logic is true (including 'not Z'). This is senseless – there is no difference between true and false anymore!

A statement-complete system is one in which every statement is either true or false, so that there is no unknowable thing. Having an incomplete system means that we can't know everything, even in principle, and that is bad. (When doing science we often think that there is just a matter of time and patience before we understand something.) This concept, I just made up, and it is a bad concept. What is truth, really, in a logical system? We still want to avoid epistemology (philosophy), so what then is truth? It is something that can be proved by using the axioms and logical rules.

In Logic/Mathematics we desire proofs. If something is true (say 'P'), then we want there to exist a string of logical arguments showing that 'P' is true. If something is true, but there is no way of knowing that it is true, that is bad. Our definition of truth in a logical system is that which can be proved within the system. A system where every statement (which can be constructed in the system) is either provably true, or provably false, is known as a complete system. Wikipedia calls this syntactically complete, and gives a nice reformulation: A system is complete if and only if "no unprovable axiom can be added to it as an axiom without introducing an inconsistency."

Let me be clear: We will only use one concept, so that truth and provability is the same thing in Logic. It is possible to make a distinction between the two, but I don't want to do that.

What Gödel tellsus is that we can never have a complete system (given that it includes basic number theory and some extra technicalities). If humanity some day decides on a logical system to use for all eternity, then either that system is inconsistent or incomplete according to Gödel's proof. In other words, since we require our system to be consistent, there will be statements that are neither true nor false in our system.

What do we call these statements that are neither true or false in our logical system? We call them independent or undecidable. Examples of these are the axiom of choice (independent in ZF-logic), and the continuum hypothesis (independent in ZFC-logic). These examples are (to mathematicians) interesting statements, so the problem that Gödel found is not some weird technicality, but something we have to deal with.

What can be done with an independent statement? We can choose to add it as an axiom, or we can choose to add its negation as an axiom. Logic does not care which we choose, it will still be consistent regardless of our choice! If we want to model observable reality, however, we might care about which is "true", as in which choice results in the best model of reality.

When we have added our axiom (or its negation), then our logic is a new and more powerful logic, where more statements are provable (also known as 'theorems'), and where more statements can be constructed. Again Gödel's proof works, and we can find new (and possibly interesting) statements that are independent in this new logical system. And so on, and so on.

Here, those who are familiar with mathematics might want to try to cheat, and add an infinite sequence of axioms, each based on a previous level's known independent statements. Even this (and generalizations of this) will not work at all. The logic you end up with is either incomplete or inconsistent.

Gödels general examples that always work in a logical system T (not true nor false):

(A Gödel number is a proof translated to a number in number theory.)

"There is no Gödel number to this statement using the logical rules of system T"

(i.e.: There is no proof of this statement.)

"The logical system T is complete"

Solving Conundrums Part 1

[I was a little unsatisfied with my last post, the second part of Q.M. logic, so to the next theme I will take a different approach; starting with a problem to solve, and, within two weeks, give hints, and within four weeks, the solutions.]

I have a conundrum for you (that word tastes like soft thunder rolling over the horizon on a warm summer day). Well, I have several. All in the form of seemingly innocent questions, that soon become quite frustrating logical puzzles. I promise that I will solve all of them in a very concrete way; I am, after all, a mathematician (I do not, however, promise that you will like the solutions, as I am not a politician).

The First Conundrum

You sit down to have your exam in [logic-something-course], and get the following multiple-choice question:

"If you were to answer this question randomly what is the probability that you would be correct?

A) 25%

B) 50%

C) 0%

D) 25%

"

What is the correct answer?

The Second Conundrum

Suppose you are at a crossroads, and there are two paths, one will lead to riches, and the other to death. In the old days there were two brothers; one who always speak the truth and another who always lie. They were, of course, identical twins. The solution was to ask both of them, which road would your brother tell me to go to get riches, and then go the opposite way.

Sadly, one of the brothers was killed by an angry customer. As they were twins, noone knows who died and who still lives on. The only thing you know about the person in front of you is that he always either speaks the truth, or he always lies. What do you ask him? Will you get rich?

The Third Conundrum

Is the following statement true?

"This sentence is false."

What about the two next sentences, are any of them true?

"The next sentence is false.

The previous sentence is true."

The Fourth Conundrum

Once upon a time, there was a proud king. His throne was usurped by a maniac, and the king was to be executed. The maniac said: "I will execute you this week, on Tuesday, Wednesday, Thursday or Friday. I will come and get you early in the morning, and it will surprise you!" The proud king then answered: "Well, fool, you cannot kill me on Friday, because then I will know it Thursday evening, so it will not be a surprise! Since you are unable to kill me come Friday, on Wednesday evening I will know it if you plan to kill me on Thursday, hence you cannot kill me on Thursday. Now, only Tuesday and Wednesday remains. So if I am not executed on Tuesday, I will know that you plan to kill me on Wednesday. Hence only Tuesday remains. But I know this, so there are no possible day when you can kill me!"

The maniac thought about this for a while, then answered "We will see". On Wednesday, the proud king was, to his surprise, executed (he had, after all, predicted that he would not be executed). Where was the flaw in his logic?

The Cat That Killed De Morgan

According to google (number of hits), it's supposed to be "the cat who killed", does anyone know for sure?

• Part two of Quantum Mechanical Logic. (Part one here).

In logic (classical/ordinary logic) we have something known as De Morgan's laws:

Let P be the statement 'It's raining outside', or any logical statement

Let Q be the statement 'The sun is shining', or any other logical statement

Then saying 'not(P and Q)' is the same as saying '(not P) or (not Q)', in words:

'It can't be both raining outside and sunny' is the same as

'It isn't raining outside, or It isn't sunny'

(yes, none of the sentences are necessarily true, but they are equivalent)

(in logic, 'A or B' means 'either A, or B, or both')

Does this work in Quantum Mechanics? No. Remember our perfectly grey cat. If you ask the cat

P: Are you black

Q: Are you grey

The statement 'not(P and Q)' [not both black and grey] is true. In the Quantum Mechanical sense, you cannot observe that it's grey and black at the same time (that is not a legal outcome). Hence (grey and black) is false, and 'not(grey and black)' is true.

The statement '(not P) or (not Q)' [not black or not grey] is true in 50% of the experiments. The part (not grey) is never true, as it starts out with being perfectly grey, and you cannot then observe it to be not grey. The statement 'not black', however, is true 50% of the time as it is a 50% chance that when we measure it to be 'not black' the cat will spontaneously become white. In total '(not P) or (not Q)' is true 50% of the time, and false the other 50% of the time.

So in Quantum Mechanical logic De Morgan's laws are not valid, the two statements are not equivalent. But in logic we require this law, so what to do? Well, even though I have called it Quantum Mechanical logic, it isn't really logic, but something else that has a strong similarity to ordinary logic; and it has some differences as we just saw.

To be fair to any mathematically inclined readers I want to add a comment. The "logical" 'or'-statement in Q.M. is usually taken to be a join of subspaces (the least subspace containing both the subspaces of Q and P), instead of a 'or' between two experimental outcomes, so that in our example the most natural thing to say is that (not black) or (not grey) is the linear span of the two, namely the whole 2-dimensional black/white subspace. Observing whether it is in this subspace would give a 'yes' with a 100% probability. It was, after all, grey. This seems to make my point moot, but alas, even with this more refined notion of "logical 'or'" you can find contradictions to De Morgan's laws (where meet and complement of subspaces is not the same as complement and join), see for example this page.

The Color Of A Cat

I'm late for my two week-appointment with my blog, so here's something special.

- How logic in Quantum Mechanics differs from the 'real world'.

Today I want to tell you one of the big secrets of Quantum Mechanics, using a parallel with a cat in it. By the time you have read this page (a couple of times) ordinary Quantum Mechanics will hopefully be clear, if not, don't hesitate to ask. Schrødinger's cat is another well known parallel with a cat, but it is about something else (in Q.M.).

We all use probability in our daily life, it's a handy tool. There is 1/6 chance of a die landing on a 6, there is 1/2 chance of a slice of bread landing with the peanut-butter-side down on the floor. Here, common sense dictates several wrong claims, like getting a 6 two times on a row (on a die) makes a third 6 less probable. Forgetting those fallacies, we all think that having enough information removes the probability. If I know the exact speed(s), air currents and form of the die and the table, I could (in theory) predict exactly on which side it would land.

So the classical world ('real world', 'everyday world') probabilities are really hidden variables. Stuff we don't know. Probability is in the map and not in theterritory.

How does this differ from Q.M.? Let us give a parallel.

Say you have a perfectly gray cat. Perfect in the sense that it is exactly halfway between white and black on your gray-scale. If you ask 'is the cat gray?' what happens? The answer is 'yes', and, of course, the cat doesn't care. If you ask 'is the cat black' what happens? You get the answer 'sort of' or 'halfway black', and, again, the cat doesn't care.

Let us assume this cat is an electron, and color is some property of that electron. The cat is still perfectly gray. If you ask 'is the cat gray?' what happens? Well, the answer is 'yes' and the cat doesn't care. It's the same, so no surprises yet! If you ask 'is the cat black?', two things can happen:

1. Answer: 'yes, black' and the cat instantly changes color to black.
2. Answer: 'no, not black' and the cat becomes non-black, which, in this case (starting with a grey cat) would actually give you a white cat.

Poor cat. But which of the answers do you get? If you had 1000 such cats and asked them all, you would get answer 1) about 500 times, and answer 2) about 500 times, so we say that the probability of getting 1) is 1/2 and same for 2).

To digress, what Schrødinger's cat is about (if I understand it correctly), is whether this is actual probability. Are there any hidden variables determining which of the cats come out black, or is there an inherent True Probability in Nature? 'God does not throw dice' -Einstein. If anyone cares, I believe Einstein to be wrong about this, and that these experimental outcomes are determined by probability. I also believe the Schrødinger's cat experiment to be a bad argument, as the cat would measure whether it was alive or dead. You don't have to be a person to do an 'experiment', and not a cat either; any two molecules on a collision course will do an experiment to see whether they collide or not.

What is special about Q.M. logic? Grey can be a 'superpositon' of white and black. How do we model this? There is a certain thing in mathematics called a Hilbert space, where colors are unit vectors (or subspaces), and a vector [1,1] can be viewed as a superposition of [1,0] and [0,1].

Why? Well, experiments show that... But why? This borders on philosophy. From a scientific point of view, this is our best model – it works (there's a friggin' flag on the Moon and a rover on Mars).