
Hyperreal numbers: infinities and infinitesimals - mmastrac
https://plus.google.com/117663015413546257905/posts/14b9fdM62un
======
4ad
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.

[1]
[http://en.wikipedia.org/wiki/Constructivism_(mathematics)](http://en.wikipedia.org/wiki/Constructivism_\(mathematics\))

[2]
[http://en.wikipedia.org/wiki/Intuitionistic_logic](http://en.wikipedia.org/wiki/Intuitionistic_logic)

[3]
[http://en.wikipedia.org/wiki/Synthetic_differential_geometry](http://en.wikipedia.org/wiki/Synthetic_differential_geometry)

[4]
[http://en.wikipedia.org/wiki/Dual_number](http://en.wikipedia.org/wiki/Dual_number)

[5] [http://math.andrej.com/2008/08/13/intuitionistic-
mathematics...](http://math.andrej.com/2008/08/13/intuitionistic-mathematics-
for-physics/)

[6]
[http://home.imf.au.dk/kock/sdg99.pdf](http://home.imf.au.dk/kock/sdg99.pdf)

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cgs1019
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.

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mcguire
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](http://en.wikipedia.org/wiki/Calculus_Made_Easy)

[2] [http://www.amazon.com/Calculus-Made-Easy-Silvanus-
Thompson/d...](http://www.amazon.com/Calculus-Made-Easy-Silvanus-
Thompson/dp/0312185480)

[3]
[http://www.gutenberg.org/ebooks/33283](http://www.gutenberg.org/ebooks/33283)

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arh68
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/Dedekind_cut)
[http://en.wikipedia.org/wiki/Surreal_number](http://en.wikipedia.org/wiki/Surreal_number)

~~~
VLM
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.

~~~
malka
Such a clickbait title... Knuth was really ahead of his time.

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mkl
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](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](http://www.math.wisc.edu/~keisler/foundations.html)

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Grue3
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...](http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem)

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Sephiroth87
It's been a while, but I seem to remember that this is actually how I was
thought calculus...

~~~
analog31
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.

~~~
mcguire
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.

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MisterMashable
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.

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ianopolous
I also wrote a blog post about these numbers recently.
[http://ianopolous.github.io/maths/surreal/](http://ianopolous.github.io/maths/surreal/)

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protonfish
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](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.)

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darkxanthos
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](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
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.

~~~
mcguire
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](http://en.wikipedia.org/wiki/George_Berkeley)

[2]
[http://en.wikipedia.org/wiki/The_Analyst](http://en.wikipedia.org/wiki/The_Analyst)

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mathgenius
This looks alot like (forward) automatic differentiation.

~~~
fdej
Yes, it's basically dual numbers (or power series -- when considering inverses
of infinitesimals, you get Laurent series) with different terminology.

~~~
4ad
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.

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silentvoice
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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hga
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...](http://www.amazon.com/Infinitesimal-Calculus-Dover-Books-
Mathematics/dp/0486428869/) It's short and focused.

------
EGreg
For those interested in investigating a larger concept of "hyperreal" numbers:
[http://math.stackexchange.com/questions/221334/whats-the-
dif...](http://math.stackexchange.com/questions/221334/whats-the-difference-
between-hyperreal-and-surreal-numbers)

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fxn
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](http://advogato.org/person/fxn/diary/475.html).

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doctorpangloss
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.

~~~
noelwelsh
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.

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ChaoticGood
John Baez post - love this guys work. I am striving to learn enough math to
fully appreciate his posts.

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PaulHoule
Either way I don't believe in the axiom of choice.

~~~
kazinator
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.

~~~
PaulHoule
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.

