Some concepts are intuitive and weird means they are not incomprehensible.It means that you have not found a way to make sense out of it.It does means they can’t be understood by any people.Things which may seems simple are not really simple.If it seems simple means you have not looked at in different directions.When you look at the same problem in different directions you can see some interesting and intricating ideas.This is true for learning also.The ability may differ for some people but everyone is entering into the sea .
Is normal things are normal ,similarly abnormal things are abnormal ?Ans:The answer is No.Humans are always fooled by their intuitions(this answer will make sense at the end) Quantum mechanics is a kind of physics where people go at atom level and they observe something with the help of it they are predicting something related to the universe.Their predictions are much more counterintuitive when they keep on going much more smaller level.But is this really valid?.This is the thing I am going to explain with a simple example of numbers concept.
Oh! Sorry for saying the word simple for numbers.Because in reality we know numbers are not simple as we think.These are not just digits .They are real valid objects.Since you have no identity on your own.You are recognized by the society by the associated number you have like social security .they have just 10 digits but how they are able to rule the world compared to the alphabets of the language.English has 26 alphabets but japanese and some other languages have more than 6000 or more alphabets .why they didn’t win the battle with the numbers .The Answer is very simple .Actually I have told you the answer .The answer is the question itself.Isn’t it funny?.since in order to say that itself I have used an associated numbers .So everything in the world can be just decrypted by numbers.
Why quantum mechanics makes sense.I am going to explain with the philosophical concept associated with the number theory concept called cantor set.Since when infinity enters the battle philosophical things will come into action.okay lets get into action.
Cardinality is the number of elements in the given set .
e.g consider a set A={1,2,3} ,B={a,b,c} .The cardinality of A is 3 and that of B is 3.
Cantor said that the cardinality of these two sets {1,2,3,4,5,………………….} and {2,4,6,8,10,12,14……………} are equal.
Similarly we can say that {1,2,3,4,5……….} and {1000,1001,1002,1003,1004,…………..} are equal .So we can find the numerous examples like this when infinity enters the battle . Why the cardinality of the following sets{1,2,3,4,5…………….} and {2,4,6,8,10………} and {100,101,102,103………} and {1000,1001,1002,1003……..} are equal?
Are they really equal?What is happening at the backend.
Answers are simple and interesting but before that we need some prior definitions.
Discrete values: Anything that can be listed are called discrete values.e.g integers,rational numbers etc…so when we say integers we are looking at a isolated points on the real number lines . similarly for rational numbers case but you have to go depth in order to understand it.Since when you consider an interval example[0,1] the middle point gives another rational numbers for an infinite times .It does not means that they are completely dominating on the real line.But there are numbers called irrational numbers which are like microorganisms which are really dominating the real number line.They are the rulers of the real number line.Consider some basic biology terms dominant and recessive for fun.so that you can appreciate the beauty behind it.
Continuous values:Anything that can’t be listed are called continuous values.e.g [0,1],[1,2],R etc….but in case of continuous values you have never able to list them even if you take your full lifetime or the time of universe.You can’t even able to list the numbers within an interval within[0,1] itself.The cardinality of R and [0,1] are same.
If you know the limit concepts in calculus then recall it otherwise leave it .This will help you to make much more sense.But I will do it with simple examples than these intuitive concepts.
So we can see that natural numbers ,integers,rational numbers can also be listed so they are called as countable set .In simple words anything that is discrete can be listed so that they can be paired with others sets that are listed like these.why I told you to recall the limit concepts some things can be listed easily some requires little intelligence.
First I will explain in a cantor manner .And then in a childish way by me.Cantor said that any two sets that can be listed with another are countable .The two sets that can’t listed with one another are called uncountable(means there are different size of infinity)
So we can see natural numbers are countable.Similarly for integers
We got the answer with the simple logic called oscillation between positive and negative numbers .So we can list now .so they are countable.similarly this holds true for rational numbers .
But now we got he question are all sets are countable no it is not consider a set between [0,1] with natural numbers .you can’t do a mapping like the prevous ones .why since you can never can list all the the numbers between 0 and 1. Cantor explain this with this simple cantor diagonalization proof which is very simple.
Suppose you have listed the three numbers(consider 3 for example)0.255356….0.34555891…0.6589…….
These are the numbers that you have listed and you are saying that you have listed all the numbers .then I can create a new number which is not in your list by looking at a diagonal unit matrix 0.357…..(3 replaced for first decimal no in first no and 5 for second decimal no in second no and 7 for third decimal no in third no).
Now my childish explanation consider a sphere of radius (conisder for example laddu)with small size and big sizes . now you can list them in pairs with one to one
In this way when you are pairing you do not give matter to big sizes mapped with small sizes ..Also it is discrete you are able to list them.
Now for funny visualization put into grinder and make a powder as possible as you can do now you can see that you are not able to list them so one-to-one matching is not possible .Also one more thing we can observe that sphere with big radius will obviously provide more powder compared to sphere with small radius.In this way we are unintentionally giving importance to sizes.This leads to a conclusion that there are sets which are not countable and also among them there are bigger sizes of infinity are there.why you are not able to list them because they are continuous.
Also this can be done with the simple graphically
e.g for {1,2,3,4……} and {2,4,6,8….} y=2x functions but defined only for isolated points
Which is a straight line for any given x value you can always find an corresponding y values.Similarly for {1,2,3,4……….} and {….,-4,-3,-2,-1,0,1,2,3,………} are as follows
When you look at graphically it will look like this .So now we can easily verify the cantor concept with a simple test called horizontal line test.
Horizontal line test:To determine whether the function is a one-to-one.Draw a horizontal at each solutions whether any other one more solution has been found on the horizontal line drawn.In this case and previous case which is no way possible so these are one-to-one function.
Before arriving to the conclusion one more simple example to understand it very deeply .why the word one-to-one function is very important .what role does it plays in the cantor’s concept.
Note:It is very important to analyze the defintion from nook and corner .Each word you consider unimportant may play a crucial role like the word here one-to-one.How depth you are going that is important that can decides your level of understanding of that concept especially this is true in case of mathematics.For you 0.9,0.99,0.999,0.9999..... all these may be equal but for a mathematician it is not.
you may leave it and put the number like 1 but behind it whole universe is arising that you can realize it's depth when you study only analysis and calculus at a greater depth.
Consider squares and square root of a number and as well as cube and root of a number
As we can see the y=x^2 function since it is not one to one it does not cover the whole range so it’s inverse function is not defined for negative numbers where as in case of cube and cuberoot function that problem did not arise since it is one-to-one function.So it’s inverse function has values for all numbers including negative numbers.
So now we atlast arrived at the conclusion(moral of the story).the interval [0,1] is o neglible compared to the size of [-R,R] but we can get an complete idea about the universe of numbers by just taking a neglible part of it .Similarly quantum mechanics is valid .Actually the [0,1] is not a neglible part it is as same size as R .so studying the [0,1] means we are indirectly studying the entire real number line(entire universe of numbers)similarly this holds for quantum mechanics .scientists who are studying atoms and much smaller level are indirectly studying the universe.Everything in disguise.We have to come out of our ignorance in order to understand the reality.why is this because as albert Einstein said the two things are infinite. One is universe and another one is human stupidity .There is question about universe whether it is finite or not.But the second one is valid I think so.
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