I hope that you can, with your video lectures, help me understand how to make sense of the philosophical implications of quantum mechanics. As of now, I am not sure how to proceed given your stricture against the use of infinite sets and axiomatic systems. For example, how does one deal with the concept of diffeomorphism invariance between coordinate systems without Real analysis? I was hopping that you would cover Robinson's non-standard Arithmetic and that this would point me toward a solution
@POWLIHERE22 I will be talking about non-standard arithmetic at some later point in this series. It will certainly be a while before we get close to quantum mechanics!
I have just watched this for a second time and still perplexed.
It seems to me your argument is this: because the natural numbers is a very difficult collection of objects to understand, we should not think of them as a whole object.
What about finite groups? Can we consider these collections of objects as complete sets? Of course we can, but they can be extremely difficult to understand, just as difficult as large natural numbers.
Just because it is very difficult to comprehend and describe individual "extremely large" numbers does not mean we should not be able to think of the natural numbers as a completed set!
In this video you are going along just fine, convincing us that the natural numbers is indeed a very complicated collection of objects but then you use this to jump to the conclusion that the natural numbers does not exist as a set out of nowhere!
On the contrary, the reality of the difficulty of factoring large numbers is quite widely acknowledged and has important applications. The underlying reason is however not understood as due to the size of the numbers, but to the problems with understanding factoring well enough. Will you prove unique factorization?
In the first or second video he said that it is very easy to understand the natural numbers, they are just strokes on paper. Now he says that it is very difficult. So which is it?
@tommyrjensen To say what a natural number is relatively easy, at least in an informal way. But to talk about `all natural numbers'' is quite different. As I try to explain, the further you go in the natural numbers, the more complicated they get. Ultimately one is overwhelmed by the immensity of really big numbers, and arithmetic with them becomes virtually impossible. This is the reality of number theory, a reality that is rarely acknowledged!
"to talk about `all natural numbers'' is quite different"
Why is this different? There is nothing wrong with the statement that the sum of 2 natural numbers is always a natural number. (m+n is a natural number for all natural numbers m and n).
Some people are overwhelmed by a simple long division and doing arithmetic with numbers smaller then 10.
Also, The field of rational numbers is infinite. Does this cause problems as well in your opinion?
What you're presenting is a version of Ackerman's function. I once used Ackerman's function to win a contest to send an email with the biggest number we could come up with in 256 characters or less. I was able to blow my mind by contemplating it; I quickly found numbers that were just inconceivably big, that I just couldn't get a handle on.
Well, 2(*_100)3 = 2(*_99)(2(*_99)2), and it's easy to work out that 2(*_n)2 = 4 for any number n, so 2(*_100)3 = 2(*_99)4, and this is clearly less than 27(*_99)15 because both arguments are smaller, while the operator is the same.
Doron Zeilberger calls it Cantor's "paradise of fools".
"infinitarian lore is sooo boring and the Bourbakian abstract nonsense leaves you with such a bitter taste that it feels more like Hell." (Opinion 68)
"Cantor's 'paradise' as well as all modern axiomatic set theory is
based on the (self-contradictory) concept of actual infinity." (A.A. Zenkin)
Your example at the end of the video, namely the one where you ask us to compare two outrageously large numbers, is besides the point. Mathematicians don't think about size in such a way that one would have to directly compare such numbers. Size is thought about in terms of correspondence between sets.
I hope that you can, with your video lectures, help me understand how to make sense of the philosophical implications of quantum mechanics. As of now, I am not sure how to proceed given your stricture against the use of infinite sets and axiomatic systems. For example, how does one deal with the concept of diffeomorphism invariance between coordinate systems without Real analysis? I was hopping that you would cover Robinson's non-standard Arithmetic and that this would point me toward a solution
POWLIHERE22 1 month ago in playlist MathFoundations
@POWLIHERE22 I will be talking about non-standard arithmetic at some later point in this series. It will certainly be a while before we get close to quantum mechanics!
njwildberger 1 month ago
I have just watched this for a second time and still perplexed.
It seems to me your argument is this: because the natural numbers is a very difficult collection of objects to understand, we should not think of them as a whole object.
What about finite groups? Can we consider these collections of objects as complete sets? Of course we can, but they can be extremely difficult to understand, just as difficult as large natural numbers.
tothemesosphere 2 months ago
Just because it is very difficult to comprehend and describe individual "extremely large" numbers does not mean we should not be able to think of the natural numbers as a completed set!
In this video you are going along just fine, convincing us that the natural numbers is indeed a very complicated collection of objects but then you use this to jump to the conclusion that the natural numbers does not exist as a set out of nowhere!
tothemesosphere 2 months ago in playlist MathFoundations
Thinking about star minus one, minus two, etc....
So if we do it this way, do we get "extremely small numbers" perhaps?
relike868p 2 months ago
On the contrary, the reality of the difficulty of factoring large numbers is quite widely acknowledged and has important applications. The underlying reason is however not understood as due to the size of the numbers, but to the problems with understanding factoring well enough. Will you prove unique factorization?
tommyrjensen 7 months ago
In the first or second video he said that it is very easy to understand the natural numbers, they are just strokes on paper. Now he says that it is very difficult. So which is it?
tommyrjensen 7 months ago
@tommyrjensen To say what a natural number is relatively easy, at least in an informal way. But to talk about `all natural numbers'' is quite different. As I try to explain, the further you go in the natural numbers, the more complicated they get. Ultimately one is overwhelmed by the immensity of really big numbers, and arithmetic with them becomes virtually impossible. This is the reality of number theory, a reality that is rarely acknowledged!
njwildberger 7 months ago
@njwildberger
"to talk about `all natural numbers'' is quite different"
Why is this different? There is nothing wrong with the statement that the sum of 2 natural numbers is always a natural number. (m+n is a natural number for all natural numbers m and n).
Some people are overwhelmed by a simple long division and doing arithmetic with numbers smaller then 10.
Also, The field of rational numbers is infinite. Does this cause problems as well in your opinion?
sftw009 5 months ago
What you're presenting is a version of Ackerman's function. I once used Ackerman's function to win a contest to send an email with the biggest number we could come up with in 256 characters or less. I was able to blow my mind by contemplating it; I quickly found numbers that were just inconceivably big, that I just couldn't get a handle on.
MistyGothis 9 months ago
anybody figure out which of the 2 numbers were bigger? I'd also be interested in how to solve for.
27(*_99)15 vs 2(*_100)3
samruby82 11 months ago
@samruby82
Well, 2(*_100)3 = 2(*_99)(2(*_99)2), and it's easy to work out that 2(*_n)2 = 4 for any number n, so 2(*_100)3 = 2(*_99)4, and this is clearly less than 27(*_99)15 because both arguments are smaller, while the operator is the same.
MooOfDoom 4 months ago
Rule of exponents for a power of a power:
For all positive integers m and n,
(a^m)^n = a^(m*n)
To find a power of a power, you multiply the exponents.
MegaDarthraider 1 year ago
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MemeMachine1 1 year ago
Doron Zeilberger calls it Cantor's "paradise of fools".
"infinitarian lore is sooo boring and the Bourbakian abstract nonsense leaves you with such a bitter taste that it feels more like Hell." (Opinion 68)
"Cantor's 'paradise' as well as all modern axiomatic set theory is
based on the (self-contradictory) concept of actual infinity." (A.A. Zenkin)
mateo3470 1 year ago
The lecturer shouldn't mention the variable letters so much, this puts the listener to sleep.
exwaan 1 year ago
Very good video!
kalvhult 2 years ago
very poorly explained! What about some examples as you proceed with the symbolic operations?
nandoorihuelsimon 2 years ago
Comment removed
DarwinsGarden 2 years ago
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DarwinsGarden 2 years ago
This has been flagged as spam show
Your example at the end of the video, namely the one where you ask us to compare two outrageously large numbers, is besides the point. Mathematicians don't think about size in such a way that one would have to directly compare such numbers. Size is thought about in terms of correspondence between sets.
DarwinsGarden 2 years ago
Comment removed
DarwinsGarden 2 years ago
Ha! Graham's number comes to mind.
RSKueffner 2 years ago
"Thinking of the natural numbers as a completed set is not correct mathematics", can I quote that in my MATH3611 homework :)
prunsunk 2 years ago
I think you should!
njwildberger 2 years ago
i'd love to see a method to compare the two large numbers mentioned at the end of the vid. thanks!
TanvirKaykobad 2 years ago
The second of those two numbers is not so intractible. However if the 3 was changed to a four, it would be much harder.
njwildberger 2 years ago
holy...
rubixcubesolve 2 years ago