The (in)famous rallying cry of Jean Dieudonné is the title of a new blog by “Nick Bornak” (hmmm, I wonder where the name came from…). The author, a student of mathematics and philosophy, started the blog in order to have a place to write about mathematics (for that, WordPress is pretty much the only way to go at the moment).

One of his major interests is nonstandard analysis, the very concept of which I used to despise until I read this brilliant post by Terence Tao. If a Fields Medalist says it’s okay, then it must be okay! Actually, what Tao’s post helped me to realize was that so-called “nonstandard analysis” can be regarded as just another item in the “soft” vocabulary of (“standard”) analysis, like asymptotic notation, or the definition of continuity in terms of inverse images of open sets. Since I absolutely adore “soft” mathematics (to the point where one of my missions in life is to “mollify” as much “hard” mathematics as I possibly can), you can see why I would find this point of view so appealing!

Before this, I used to see nonstandard analysis as, at best, an obfuscatory way of formulating ordinary analysis; or at worst, a sort of contrary position in the ontology/epistemology of mathematics, like constructivism (except perhaps on the “other extreme”). The way nonstandard analysis was described — as a way of making Leibniz’s infinitesimal reasoning rigorous — irritated me to no end. Didn’t these people realize that the work of the likes of Cauchy, Weierstrass, and Cantor had already made that reasoning rigorous? Not to understand this — simply because of Leibniz’s “different” vocabulary — is to exhibit the sort of over-concrete literal-mindedness that is all too typical of certain historians — particularly historians of mathematics — and, for that matter, music theorists. It thus seemed as if (not for the first time, alas) a whole field had been founded on a simple lack of intellectual agility!

Luckily, thanks largely to Terry Tao, I’ve since moved beyond this. Perhaps Nick Bornak will over time be able to still further illuminate the virtues of the “nonstandard” way of thinking. (That word “nonstandard” really is a turn-off, I have to say — to me it carries a suggestion of crackpottery.)

(Incidentally, though I’m a great admirer of Dieudonné, I don’t have anything against triangles. In fact, I think I could discuss them in a way that he would approve of. But that’s for another occasion…)

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One Response to Death to triangles! (and life to ultrafilters…)

Because of the way the non-standard number system
once appeared (quite unexpectedly) in my research,
I’ve arrived at a more algebraic point of view about
non-standard numbers which differs in outlook from
managing epsilons or making infintessimal arguments
rigorous or the standpoint of logical model theory.
Just to be clear, I am in no way claiming that
this point of view is better (or worse) than the
others, but rather offer it with the thought that
seeing the same subject from different perspectives
can lead to deeper understanding and that seeing
the same mathematical structure arise in several
contexts can suggest that it is something
interesting which is worthy of further examination.

A well-known characterization of the real number
system is as the unique complete ordered field (up
to isomorphism). Hence, if one wants to extend
this field, one will have to give up either
ordering or completeness (or perhaps both).

If one gives up ordering, one can construct an
algebraic field extension of the real numbers,
which has the pleasant property of being
algebraically complete as well as complete in the
sense of convergence. One can then proceed
to extend the operations of analysis to functions
over this larger field and finds that, even when
one is only interested in some question involving
real numbers (say deriving trigonometric
identities or computing a definite integral),
one can often save labor by first working with
complex analysis and then taking the real part
at the end of the day. (As Watson once quipped,
the shortest path between two real facts often
passes through the complex plane.)

Suppose instead that one wants to extend the
real numbers to a larger ordered field. One
then needs to give up completeness and arrives
at non-standard analysis. The epsilons arise
naturally — one statement of completeness is
Archimides principle, the negation of which is
tantamount to the assertion that the field
contains infinitessimal elements.

The basic methods of non-standard analysis are
reminiscent to me of complex analysis. Just as
in the latter, one analytically extends real
analytic functions to complex functions in a
unique manner so too, in the latter, one extends
real functions defined in first order logic
to non-standard functions in a unique fashion.
Just as the principle of persistence of relations
allows one to carry over functional equations and
differential equations to the complex plane, so
too does the transfer principle work.
Just as one might take the real part at the end
of some complex operations, so too one often
takes the standard part after some non-standard
operations to obtain a result in good old
real analysis.

As for the ultrafilters, they arise from looking
at limits of sequences algebraically. (I learned
this from Ebbinghaus’ wonderful book on numbers.)
When talking about taking limits of sequences,
what we are considering is assigning numbers to
sequences is some reasonable way. As for what the
term “reasonable” might denote in the last
sentence, we presumably don’t want to assign
numbers to sequences haphazardly, but in a way
which is consistent with the structure; at the
least, we might want the limit of a sum of sequences
to be the sum of limits and likewise the limit of
products to be the product of limits, for instance.

Algebraically, this desideratum about sums and
products says that the limit operation should be
a morphism from the ring of sequences to that of
real numbers. Now, in real analysis, we only
obtain such a morphism for the proper subalgebra
of convergent sequences; let’s see if we can do
better by picking a different field as target of
the morphism. A basic result of commutative algebra
states that a morphism from a ring to a field is
isomorphic to quotienting by a maximal ideal.
Hence, we should llok for the maximal ideals of
the ring of real sequences. A little work shows
that these are in one-to-one correspondence with
the maximal ideals of the Boolean algebra of
subsets of the integers, which are trivially
related to ultrafilters. This too, is a parallel
to complex analysis — just as one can obtain
complex numbers by extending the real number
system so as to make every polynomial have a
root, so too one can obtain the non-standard
nunmber system as an extension in which every
sequence has a limit.

By the way, I am no more put off by the term
“non-standard” than by the term “imaginary”
(which could suggest that complex analysts are
dayreamers) or the term “degenerate” or the
term “irrational” — mathematicians have a
habit of using negative, even pejorative terms
to describe new concepts which challenge
naive intuition.

Because of the way the non-standard number system

once appeared (quite unexpectedly) in my research,

I’ve arrived at a more algebraic point of view about

non-standard numbers which differs in outlook from

managing epsilons or making infintessimal arguments

rigorous or the standpoint of logical model theory.

Just to be clear, I am in no way claiming that

this point of view is better (or worse) than the

others, but rather offer it with the thought that

seeing the same subject from different perspectives

can lead to deeper understanding and that seeing

the same mathematical structure arise in several

contexts can suggest that it is something

interesting which is worthy of further examination.

A well-known characterization of the real number

system is as the unique complete ordered field (up

to isomorphism). Hence, if one wants to extend

this field, one will have to give up either

ordering or completeness (or perhaps both).

If one gives up ordering, one can construct an

algebraic field extension of the real numbers,

which has the pleasant property of being

algebraically complete as well as complete in the

sense of convergence. One can then proceed

to extend the operations of analysis to functions

over this larger field and finds that, even when

one is only interested in some question involving

real numbers (say deriving trigonometric

identities or computing a definite integral),

one can often save labor by first working with

complex analysis and then taking the real part

at the end of the day. (As Watson once quipped,

the shortest path between two real facts often

passes through the complex plane.)

Suppose instead that one wants to extend the

real numbers to a larger ordered field. One

then needs to give up completeness and arrives

at non-standard analysis. The epsilons arise

naturally — one statement of completeness is

Archimides principle, the negation of which is

tantamount to the assertion that the field

contains infinitessimal elements.

The basic methods of non-standard analysis are

reminiscent to me of complex analysis. Just as

in the latter, one analytically extends real

analytic functions to complex functions in a

unique manner so too, in the latter, one extends

real functions defined in first order logic

to non-standard functions in a unique fashion.

Just as the principle of persistence of relations

allows one to carry over functional equations and

differential equations to the complex plane, so

too does the transfer principle work.

Just as one might take the real part at the end

of some complex operations, so too one often

takes the standard part after some non-standard

operations to obtain a result in good old

real analysis.

As for the ultrafilters, they arise from looking

at limits of sequences algebraically. (I learned

this from Ebbinghaus’ wonderful book on numbers.)

When talking about taking limits of sequences,

what we are considering is assigning numbers to

sequences is some reasonable way. As for what the

term “reasonable” might denote in the last

sentence, we presumably don’t want to assign

numbers to sequences haphazardly, but in a way

which is consistent with the structure; at the

least, we might want the limit of a sum of sequences

to be the sum of limits and likewise the limit of

products to be the product of limits, for instance.

Algebraically, this desideratum about sums and

products says that the limit operation should be

a morphism from the ring of sequences to that of

real numbers. Now, in real analysis, we only

obtain such a morphism for the proper subalgebra

of convergent sequences; let’s see if we can do

better by picking a different field as target of

the morphism. A basic result of commutative algebra

states that a morphism from a ring to a field is

isomorphic to quotienting by a maximal ideal.

Hence, we should llok for the maximal ideals of

the ring of real sequences. A little work shows

that these are in one-to-one correspondence with

the maximal ideals of the Boolean algebra of

subsets of the integers, which are trivially

related to ultrafilters. This too, is a parallel

to complex analysis — just as one can obtain

complex numbers by extending the real number

system so as to make every polynomial have a

root, so too one can obtain the non-standard

nunmber system as an extension in which every

sequence has a limit.

By the way, I am no more put off by the term

“non-standard” than by the term “imaginary”

(which could suggest that complex analysts are

dayreamers) or the term “degenerate” or the

term “irrational” — mathematicians have a

habit of using negative, even pejorative terms

to describe new concepts which challenge

naive intuition.