There are many different rigorous definitions of infinitesimals. Many don't require creating* any new sets, though many do require intuitionist logic[2]. My favorite are nilpotent infinitesimals which give rise to synthetic differential geometry[3] and dual numbers[4]. Compared to hyperreal infinitesimals, nilpotent infinitesimals are much easier to construct.
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.
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."
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. :)
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
>>> 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.
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.
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.)
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.
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.
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.
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.
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.
[+] [-] 4ad|11 years ago|reply
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
[+] [-] cgs1019|11 years ago|reply
[+] [-] mcguire|11 years ago|reply
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
[+] [-] arh68|11 years ago|reply
[1] http://en.wikipedia.org/wiki/Dedekind_cut http://en.wikipedia.org/wiki/Surreal_number
[+] [-] VLM|11 years ago|reply
"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.
[+] [-] mkl|11 years ago|reply
Keisler also has another shorter book on the same stuff: http://www.math.wisc.edu/~keisler/foundations.html
[+] [-] Grue3|11 years ago|reply
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...
[+] [-] Sephiroth87|11 years ago|reply
[+] [-] analog31|11 years ago|reply
>>> 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.
[+] [-] MisterMashable|11 years ago|reply
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.
[+] [-] ianopolous|11 years ago|reply
[+] [-] protonfish|11 years ago|reply
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.)
[+] [-] darkxanthos|11 years ago|reply
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.
[+] [-] pfortuny|11 years ago|reply
Evanescent quantities are quite natural to me.
[+] [-] mathgenius|11 years ago|reply
[+] [-] fdej|11 years ago|reply
[+] [-] silentvoice|11 years ago|reply
[+] [-] unknown|11 years ago|reply
[deleted]
[+] [-] hga|11 years ago|reply
[+] [-] EGreg|11 years ago|reply
[+] [-] fxn|11 years ago|reply
[+] [-] doctorpangloss|11 years ago|reply
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.
[+] [-] noelwelsh|11 years ago|reply
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.
[+] [-] ChaoticGood|11 years ago|reply
[+] [-] PaulHoule|11 years ago|reply
[+] [-] kazinator|11 years ago|reply