27 Jun 16
I just watched this [/youtube] video about surreal numbers and also checked the Wikipedia page:
https://en.wikipedia.org/wiki/Surreal_number
It seems to claim that there exists numbers smaller than the smallest real number or larger than infinity, division by infinity etc. It all seems like nonsense to me.
27 Jun 16
Originally posted by twhiteheadAs I get it it's just the real numbers, but with infinitesimals added between zero and the positive reals and some infinite numbers added on the right of the set of reals. I think it can be made consistent, which is the interest to mathematicians, but I think they are more a mathematical curiosity than a useful calculational tool. From a pure maths point of view I think the interest is simply whether a number system with infinitesimals can be made consistent. The Wikipedia page is reasonably good.
I just watched this [youtube]https://www.youtube.com/watch?v=mPn2AdMH7UQ[/youtube] video about surreal numbers and also checked the Wikipedia page:
https://en.wikipedia.org/wiki/Surreal_number
It seems to claim that there exists numbers smaller than the smallest real number or larger than infinity, division by infinity etc. It all seems like nonsense to me.
https://en.wikipedia.org/wiki/Surreal_number
27 Jun 16
8 edits
Originally posted by DeepThoughtI noticed this quote in the hyperlink in that link:
As I get it it's just the real numbers, but with infinitesimals added between zero and the positive reals and some infinite numbers added on the right of the set of reals. I think it can be made consistent, which is the interest to mathematicians, but I think they are more a mathematical curiosity than a useful calculational tool. From a pure maths poi ...[text shortened]... sistent. The Wikipedia page is reasonably good.
https://en.wikipedia.org/wiki/Surreal_number
https://en.wikipedia.org/wiki/Infinitesimal
" ... the followers of Cantor, Dedekind, and Weierstrass sought to rid analysis of infinitesimals, and their philosophical allies like Bertrand Russell and Rudolf Carnap declared that infinitesimals are pseudoconcepts, ..."
although many other philosophers would disagree with that.
I wonder who is right? I can't tell without a huge amount of proper research into infinitesimals which I haven't done but my intuition has always made me highly suspicions of the concept of infinitesimals (a number between zero and the reciprocal of any positive integer no matter how large? That seems like nonsense to me ) even though, strangely, I have heard of infinitesimals often been mathematically used with apparently correct results.
Anyone with an opinion on this? Are infinitesimals 'pseudoconcepts' ?
28 Jun 16
Originally posted by humyBut you can add an infintesimal to any number, I mean 4 + 0.00000000000000000000001 is a real number so why couldn't you have an infinite number of zero's to the real number at the end of the string?
I noticed this quote in the hyperlink in that link:
https://en.wikipedia.org/wiki/Infinitesimal
" ... the followers of Cantor, Dedekind, and Weierstrass sought to rid analysis of infinitesimals, and their philosophical allies like Bertrand Russell and Rudolf Carnap declared that infinitesimals are pseudoconcepts, ..."
although many other philosophers wou ...[text shortened]... rently correct results.
Anyone with an opinion on this? Are infinitesimals 'pseudoconcepts' ?
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28 Jun 16
Originally posted by humyhttp://math.oregonstate.edu/bridge/papers/differentials.pdf
I noticed this quote in the hyperlink in that link:
https://en.wikipedia.org/wiki/Infinitesimal
" ... the followers of Cantor, Dedekind, and Weierstrass sought to rid analysis of infinitesimals, and their philosophical allies like Bertrand Russell and Rudolf Carnap declared that infinitesimals are pseudoconcepts, ..."
although many other philosophers wou ...[text shortened]... rently correct results.
Anyone with an opinion on this? Are infinitesimals 'pseudoconcepts' ?
28 Jun 16
Originally posted by humyMy feeling is that either a system is consistent or it is not. If it isn't then it's useless for any purpose. If it is consistent then it may not add anything useful to applied mathematics. If space-time turns out to be discrete are we to regard the set of real numbers as "proved wrong". I don't think a mathematical object has to have a counterpart in nature to be regarded as valid.
I noticed this quote in the hyperlink in that link:
https://en.wikipedia.org/wiki/Infinitesimal
" ... the followers of Cantor, Dedekind, and Weierstrass sought to rid analysis of infinitesimals, and their philosophical allies like Bertrand Russell and Rudolf Carnap declared that infinitesimals are pseudoconcepts, ..."
although many other philosophers wou ...[text shortened]... rently correct results.
Anyone with an opinion on this? Are infinitesimals 'pseudoconcepts' ?
28 Jun 16
7 edits
Originally posted by DeepThoughtmy feeling on that is exactly the same as you just said there. And judging purely by all the quotes I have just read that contain the word "infinitesimal" in the link just provided by joe shmo, I think our intuitions might be right. I really wish I could be much more certain though specifically because me knowing whether infinitesimal can add anything useful to applied mathematics is of some not totally insignificant importance to my current research into probability; more specifically, regarding probability density functions and what is their valid literal philosophical meaning as you consider ever smaller integrals under the curve of the function; I would like to prove infinitesimals are totally philosophically meaningless (but not necessarily meaningless in purely mathematics terms ignoring philosophical meaning) in that specific narrow context.
My feeling is that either a system is consistent or it is not. If it isn't then it's useless for any purpose. If it is consistent then it may not add anything useful to applied mathematics. ...
28 Jun 16
Originally posted by humyActually that particular case is a known mathematical fact. Any reciprocal of any positive integer, no matter how large, is a rational number. Zero is also a rational number. It is a mathematical fact that between any two distinct rationals there is always at least one real number.
my intuition has always made me highly suspicions of the concept of infinitesimals (a number between zero and the reciprocal of any positive integer no matter how large? That seems like nonsense to me )
What I find suspicious is the claim that such a number is not a real number.
28 Jun 16
Originally posted by sonhousewouldn't 4.0000....(on for infinitum) simply equal 4 ?
But you can add an infintesimal to any number, I mean 4 + 0.00000000000000000000001 is a real number so why couldn't you have an infinite number of zero's to the real number at the end of the string?
28 Jun 16
Originally posted by twhiteheadI guess what I really meant to say is;
Actually that particular case is a known mathematical fact. Any reciprocal of any positive integer, no matter how large, is a rational number. Zero is also a rational number. It is a mathematical fact that between any two distinct rationals there is always at least one real number.
...
"a number between zero and the reciprocal of an infinitely large positive integer"
which is nonsense because there is no infinitely large positive integer because infinity is not an integer.
So perhaps I should say an infinitesimal is supposed to be;
"a number between zero and the reciprocal of infinity"
anyone; is that right?
28 Jun 16
Originally posted by humyThis came up in the threads about probability. The ratio of a point in an interval to the real numbers on the interval is essentially the reciprocal of infinity. It is infinitesimal but non-zero. Without it, probability theory should be in trouble, but in reality probability theory skirts around the problem by defining probability slightly differently for intervals.
So perhaps I should say an infinitesimal is supposed to be;
"a number between zero and the reciprocal of infinity"
anyone; is that right?
I am not convinced such an infinitesimal is a number any more than infinity is and such a ratio or reciprocal cannot be treated as mathematically sound.
29 Jun 16
1 edit
Originally posted by sonhouseBut what is 'the infinitesimal'? I don't think it is possible to have an infinite number of places after the decimal and then a '1'. This would imply a greater than infinite number of places after the decimal. (we are talking about countable infinities here.)
I was talking about the infinitesimal added to the 4.00000000....