Hyperreal numbers: infinities and infinitesimals(plus.google.com)
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Hyperreal numbers: infinities and infinitesimals
https://plus.google.com/117663015413546257905/posts/14b9fdM62un
32 comments
Baez's google+ posts are always super interesting, informative and full of references for further reading. The comment threads are generally also very interesting and active. Highly recommend following him.
There's also Calculus Made Easy[1], by Silvanus Thompson, relatively recently reprinted in an edition with additions from Martin Gardner.[2] The original edition is available in the US from Project Gutenberg, though.[3]
Quoth the 'pedia: "Calculus Made Easy is a book on infinitesimal calculus originally published in 1910 by Silvanus P. Thompson, considered a classic and elegant introduction to the subject."
[1] http://en.wikipedia.org/wiki/Calculus_Made_Easy
[2] http://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/d...
[3] http://www.gutenberg.org/ebooks/33283
Quoth the 'pedia: "Calculus Made Easy is a book on infinitesimal calculus originally published in 1910 by Silvanus P. Thompson, considered a classic and elegant introduction to the subject."
[1] http://en.wikipedia.org/wiki/Calculus_Made_Easy
[2] http://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/d...
[3] http://www.gutenberg.org/ebooks/33283
Yeah, math! I like this explanation. I'm not a mathematician, but I was reading about the reals, the hyperreals, the surreals on Wikipedia last week, and the Dedekind cut [1] just seemed like a mind-blowingly simple way to look a things. I think I was taught in school how to construct the reals by Cauchy's proof. Constructing the reals used to seem like some magic trick back in the day. :)
[1] http://en.wikipedia.org/wiki/Dedekind_cut http://en.wikipedia.org/wiki/Surreal_number
[1] http://en.wikipedia.org/wiki/Dedekind_cut http://en.wikipedia.org/wiki/Surreal_number
You might get a kick out of
"Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness, 1974, ISBN 0-201-03812-9."
by Knuth. Yeah, that Knuth. I haven't read it in a decade or two but it was enjoyable and on topic.
"Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness, 1974, ISBN 0-201-03812-9."
by Knuth. Yeah, that Knuth. I haven't read it in a decade or two but it was enjoyable and on topic.
Such a clickbait title... Knuth was really ahead of his time.
This book is really good. I hope to have time to go through the whole thing in detail some day, but so far I've only read the first few chapters. If you want a (much more usable and readable) PDF version, go here: http://www.math.wisc.edu/~keisler/calc.html
Keisler also has another shorter book on the same stuff: http://www.math.wisc.edu/~keisler/foundations.html
Keisler also has another shorter book on the same stuff: http://www.math.wisc.edu/~keisler/foundations.html
If there's an infinite model of some first-order logical system, then there's a model of the same logical system of any infinite cardinality. [1]
Thus, there exists countable model of reals, as well as models of greater-than-continuum cardinalities.
[1] http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_t...
Thus, there exists countable model of reals, as well as models of greater-than-continuum cardinalities.
[1] http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_t...
It's been a while, but I seem to remember that this is actually how I was thought calculus...
That was my impression too...
>>> You can calculate the derivative, or rate of change, of a function f by doing
>>> (f(x+ε) - f(x)) / ε
>>> and then at the end throwing out terms involving ε.
... is exactly how the derivative was defined in both high school and college calc, except that the last bit was formalized by taking a limit.
Now, the concept of a limit was presented rather informally in high school, then with progressively more rigor in college calculus, and ultimately in real analysis. This approach got most of the STEM students through calculus in a finite time, but meant that only the math majors got to do calculus at a deeper level. And grad school, well, I went into physics instead. Every math course skips the nasty bits that get re-visited at a higher level later. Maybe that's just how math is.
My understanding of infinities and infinitesimals is that they are "not numbers," but are essentially a short hand notation for the processes involved in taking limits. And limits are based on sets. Granted, there may be other ways to understand the same stuff.
>>> You can calculate the derivative, or rate of change, of a function f by doing
>>> (f(x+ε) - f(x)) / ε
>>> and then at the end throwing out terms involving ε.
... is exactly how the derivative was defined in both high school and college calc, except that the last bit was formalized by taking a limit.
Now, the concept of a limit was presented rather informally in high school, then with progressively more rigor in college calculus, and ultimately in real analysis. This approach got most of the STEM students through calculus in a finite time, but meant that only the math majors got to do calculus at a deeper level. And grad school, well, I went into physics instead. Every math course skips the nasty bits that get re-visited at a higher level later. Maybe that's just how math is.
My understanding of infinities and infinitesimals is that they are "not numbers," but are essentially a short hand notation for the processes involved in taking limits. And limits are based on sets. Granted, there may be other ways to understand the same stuff.
I didn't take calculus in high school (I'm not even sure it was offered, actually. But, football!), so in my first semester of college I got hit with limits right out of the gate. I didn't know there was an alternative until much later.
"My understanding of infinities and infinitesimals is that they are 'not numbers,'"
Nope, they're perfectly good numbers. Check out the origins of "nonstandard analysis" in the mid-20th century for the one stupid trick that makes mathematicians jealous. Or something.
Now, irrationals and complexes? Them's not numbers; them's eeeeevil.
"My understanding of infinities and infinitesimals is that they are 'not numbers,'"
Nope, they're perfectly good numbers. Check out the origins of "nonstandard analysis" in the mid-20th century for the one stupid trick that makes mathematicians jealous. Or something.
Now, irrationals and complexes? Them's not numbers; them's eeeeevil.
The most fruitful definition of a real number is as a limit of a Cauchy sequence. That way is much more useful in proving theorems.
Using infinitesimals is logically valid (alternative real analysis), useful for physics and other practical calculations but not at all helpful proving theorems.
Might I add that the concept of 'nearness' introduced by Riesz is the contrapositive of the usual limit definition and might be the way real analysis is taught 100 years from now.
Hyperreals are much more involved than mere epsilontics as they include all kinds of infinities. It's so mind blowing that I simply must defer to minds like Conway to play with such things.
Using infinitesimals is logically valid (alternative real analysis), useful for physics and other practical calculations but not at all helpful proving theorems.
Might I add that the concept of 'nearness' introduced by Riesz is the contrapositive of the usual limit definition and might be the way real analysis is taught 100 years from now.
Hyperreals are much more involved than mere epsilontics as they include all kinds of infinities. It's so mind blowing that I simply must defer to minds like Conway to play with such things.
I also wrote a blog post about these numbers recently. http://ianopolous.github.io/maths/surreal/
I stumbled across a calculus textbook that used infinitesimals instead of limits and Epsilon and all that nonsense that made no sense to me when I took the official courses. https://www.math.wisc.edu/~keisler/calc.html
This made calculus actually made sense to me. I was quickly able to figure out how to take a derivative of a polynomial just from the understanding received (instead of applying memorized rules.)
This made calculus actually made sense to me. I was quickly able to figure out how to take a derivative of a polynomial just from the understanding received (instead of applying memorized rules.)
When the author says infinitesimals can be used to define calculus in a perfectly rigorous way I get a bit skeptical. After reading this Wikipedia article though I see why.
http://en.m.wikipedia.org/wiki/Hyperreal_number
I still think there's a certain elegance of not needing to define a whole new set of numbers, but there's also an elegance to the intuition of infinitesimals and infinitesimals.
http://en.m.wikipedia.org/wiki/Hyperreal_number
I still think there's a certain elegance of not needing to define a whole new set of numbers, but there's also an elegance to the intuition of infinitesimals and infinitesimals.
Actually, Newton and Leibniz were all about this: infinitesimals. It was only in the XIX Century with D'Alembert and especially when Weisrstrass was ill-digested that we got the epsilon-delta down our throats.
Evanescent quantities are quite natural to me.
Evanescent quantities are quite natural to me.
As I recall, there was a fluff-up between Bishop Berkeley[1] and the Newton camp regarding the issue[2], in the 18th century.
"Mathematical historian Judith Grabiner comments, 'Berkeley’s criticisms of the rigor of the calculus were witty, unkind, and — with respect to the mathematical practices he was criticizing — essentially correct'."
[1] http://en.wikipedia.org/wiki/George_Berkeley
[2] http://en.wikipedia.org/wiki/The_Analyst
"Mathematical historian Judith Grabiner comments, 'Berkeley’s criticisms of the rigor of the calculus were witty, unkind, and — with respect to the mathematical practices he was criticizing — essentially correct'."
[1] http://en.wikipedia.org/wiki/George_Berkeley
[2] http://en.wikipedia.org/wiki/The_Analyst
Oh totally. I know they were. From the wikipedia article though it seems like they were much less rigorous about their treatment and that it wasn't until the mid-20th century that there was a formal treatment of them.
This looks alot like (forward) automatic differentiation.
Yes, it's basically dual numbers (or power series -- when considering inverses of infinitesimals, you get Laurent series) with different terminology.
Very similar in spirit, but still different in the details. Dual numbers use nilpotent infinitesimals (which are not invertible), while non-standard (Robinson) infinitesimals are not nilpotent and are invertible.
That being said it's way easier to construct nilpotent infinitesimals using intuitionistic logic than it is to construct the hyperreals using classical logic.
That being said it's way easier to construct nilpotent infinitesimals using intuitionistic logic than it is to construct the hyperreals using classical logic.
I like the standard "epsilon/delta" approach better. In that approach the idea of "infinite" insofar as it applies to real numbers is introduced solely as a notational convenience, and is in no way necessary. All notions of limits can exist without infinity, and in this way I believe they are much more logically clear.
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One more book I found to be useful, at least for its introduction to the hyperreal number system: http://www.amazon.com/Infinitesimal-Calculus-Dover-Books-Mat... It's short and focused.
For those interested in investigating a larger concept of "hyperreal" numbers: http://math.stackexchange.com/questions/221334/whats-the-dif...
Some time ago I wrote a post with a little bit of historical background and an outline of their formal construction: http://advogato.org/person/fxn/diary/475.html.
From his article:
You can calculate the derivative, or rate of change, of a function f by doing
(f(x+ε) - f(x)) / ε
and then at the end throwing out terms involving ε
---
To be clear, this formula is typically introduced as the finite difference formula in calculus instruction.
But differentiation of polynomials is straightforward, so don't be too impressed that a simple formula finds the derivative of x-squared in his example. A bunch of simple procedures find the derivatives of polynomials.
For more complicated algebraic functions, like rational functions, nearly every calculus student is taught a collection of shortcuts that are fundamentally taking the limit of the finite difference formula as epsilon (or "h" commonly) approaches zero.
You can calculate the derivative, or rate of change, of a function f by doing
(f(x+ε) - f(x)) / ε
and then at the end throwing out terms involving ε
---
To be clear, this formula is typically introduced as the finite difference formula in calculus instruction.
But differentiation of polynomials is straightforward, so don't be too impressed that a simple formula finds the derivative of x-squared in his example. A bunch of simple procedures find the derivatives of polynomials.
For more complicated algebraic functions, like rational functions, nearly every calculus student is taught a collection of shortcuts that are fundamentally taking the limit of the finite difference formula as epsilon (or "h" commonly) approaches zero.
I think you've missed the point. You're correct that one can do basic calculus using a bunch of ad-hoc shortcuts, and it is commonly taught this way. Also, as you probably know, calculus was first formalised using limits not infinitesimals.
However this theory of infinitesimals is not ad-hoc. It is a complete and rigorous alternate formulation of calculus in terms of an extension of the reals, known as the hyperreals, that includes infinitesimals. This allows computation in a principled way that matches the intuitions on which beginning calculus is often taught.
It's pretty cool stuff, and worth reading about even if you never use it in practice.
However this theory of infinitesimals is not ad-hoc. It is a complete and rigorous alternate formulation of calculus in terms of an extension of the reals, known as the hyperreals, that includes infinitesimals. This allows computation in a principled way that matches the intuitions on which beginning calculus is often taught.
It's pretty cool stuff, and worth reading about even if you never use it in practice.
John Baez post - love this guys work. I am striving to learn enough math to fully appreciate his posts.
Either way I don't believe in the axiom of choice.
I'm with you. Anything that is required so that two spheres, each of volume V, can be produced by cutting up and re-arranging the pieces of a single sphere of volume V, is rather fishy.
Also from a constructive viewpoint you just can't do it.
Every integer or rational number has a name, and can be specified. You can even specify some transcendental numbers such as Feigenbaum's constant, e-1, pi^2 and such, but the "real" numbers which are mostly nameless and generally cannot be encountered except as part of a range, are "real" in a different way than the integers are.
Every integer or rational number has a name, and can be specified. You can even specify some transcendental numbers such as Feigenbaum's constant, e-1, pi^2 and such, but the "real" numbers which are mostly nameless and generally cannot be encountered except as part of a range, are "real" in a different way than the integers are.
Here[5] is a gentle introduction to nilpotent infinitesimals and intuitionist logic, and here[6] is a very good book on synthetic differential geometry.
* I didn't say construct, because that means something very specific in mathematics[1] and non-standard analysis is not traditionally constructive.
[1] http://en.wikipedia.org/wiki/Constructivism_(mathematics)
[2] http://en.wikipedia.org/wiki/Intuitionistic_logic
[3] http://en.wikipedia.org/wiki/Synthetic_differential_geometry
[4] http://en.wikipedia.org/wiki/Dual_number
[5] http://math.andrej.com/2008/08/13/intuitionistic-mathematics...
[6] http://home.imf.au.dk/kock/sdg99.pdf