Transcript Supplementary slides on modern infinitesimals

THE RETURN OF THE
INFINITESIMAL?
History &
Philosophy of
Calculus,
Session 10
INTRODUCTION
 So far on this course we’ve stayed close to mainstream
mathematics.
 In this final session we look at some more “fancy” material,
much of it very recent.
 These aren’t just techniques for doing complicated calculations –
they represent new ways to think about geometry and continuity, and
so they promise to raise some old philosophical questions.
 Perhaps they will stimulate new ones, too – but only if there are
philosophers who can understand them!
 We can only touch on these topics here, and we’re not experts in
them, but hopefully we can give a sense of what kind of work is being
done and why.
LOGIC AND ABSTRACTION
 Much of what’s really going on in these projects is that
seemingly concrete geometrical ideas are being translated
into the much more abstract languages of formal logic or
category theory.
 These settings tend to provide great power in exchange for
removing a lot of our intuition about what’s going on.
In some cases the purpose of the project is to prove dif ficult
things by indirect means.
 Such methods may produce results some find unsatisfying or even
illegitimate.
 In others the purpose is to generalise a successful technique
developed in one context to cover a wider range of cases.
 Roughly speaking, the more general a theory is, the weaker it is,
because it must cover many different situations. But a setting like
this can illuminate patterns and structures that are hard to see in a
more specific (and stronger) theory.
SYNTHETIC GEOMETRY
NEW GEOMETRIES
 Analytical geometry had huge successes in the 17 th and 18 th
centuries.
 Much of this was thanks to the power of calculus.
 These discoveries all took place within the world of Euclidean
geometry, which was tacitly presumed to be “the” geometry.
 In the 19 th century, new geometries were invented (or
discovered?).
 Analytic techniques were useful for investigating them; the standard
models of elliptic and hyperbolic spaces, for example, are analytic
and make ingenious use of calculus to answer such questions as:
How can we find the shortest path between two points on a curved
surface?
 But synthetic techniques also made a comeback, especially in
relation to projective geometry.
HEGEL
 In the Science of Logic , Hegel asserts that analytical geometry
has taken a wrong turn.
 http://ncatlab.org/nlab/show/synthetic%20geometry
 He is in a long tradition of metaphysicians who denied that a
continuum could be made up of points.
 The list begins with Aristotle and Eudoxus and proceeds through the
great mathematicians of the early modern period, including Newton and
Leibniz.
 It includes later figures, too, such as Poincare, Brouwer and Thom.
 These figures all felt that while points can be produced out of a
continuum by cutting, the continuum itself must be “made up of”
something else.
 But the use of infinitesimals in analysis had been shown to lead
to contradictions.
 As Bell puts it, “the proscription of infinitesimals did not succeed in eliminating
them altogether but, instead, drove them underground” – scientists, engineers
and even pure mathematicians continued to use them, but sometimes translated
their results into more “respectable” language for publication .
METAPHYSICS OR MATHS?
 It’s not always clear what’s at stake in these developments.
 Sometimes it seems the issue is that nineteenth century analysis
failed to grasp the real nature of the continuum, and that finding a
way to reintroduce infinitesimals is about “fixing” this metaphysical
failure.
 At other times, though, it looks like the issue is that calculus runs up
against certain inconvenient logical limitations that a new way of
looking at it can overcome, leading to practical breakthroughs.
 At still others, it seems we’re just experimenting with these ideas in
the spirit of pure maths: Let’s set things up this way and see if
anything interesting happens.
 It should also be said that these are niche areas: most
mathematicians working in analysis don’t use them, or even
know much about them, and they’re rarely taught to students.
TWO DIFFERENT PROJECTS
 Moerdijk & Reyes, p. v
NONSTANDARD
ANALYSIS
FOUNDATION OR WORKBENCH?
 Nonstandard analysis proposes a whole new number system that
includes infinitesimally small and infinitely large numbers.
 Altogether, these are called the hyperreal numbers.
 On the face of it, it seems that by doing this it’s offering an
alternative foundation for calculus: throw away the real numbers
with their awkward, limit -based construction and replace them
with this new model of the continuum instead!
 But this is only half true.
 Nonstandard analysis creates an alternative version of calculus that is,
in a special sense, equivalent to it.
 The main motivation is that it provides a sneaky, indirect way to
prove some tricky theorems.
 So this is more of a practical project.
 Nonstandard analysis was taught, experimentally, in some American
schools in the 1970s as part of the “New Math”. It was controversial; if
you want to see how it was done, check out Keisler’s book.
THE STRATEGY
 Since the birth of the calculus, people working with it have
used intuitive ideas about “infinitesimal quantities” to think
about what they’re doing.
 As we’ve seen, this quickly became intellectually disreputable
– philosophers like Berkeley make it seem that such notions
were nonsensical.
 Yet they did lead the creators of the calculus in the right direction –
maybe they weren’t completely the wrong idea.
 What if we could create a new number system – the hyperreal
numbers – including the real numbers but also including the
infinitesimals Leibniz and (early on) Newton had in mind?
 But we’ll have to be careful! Berkeley was right to warn of the
absurdities lurking around every corner.
 Fortunately, modern logic enables us to be very careful when we
need to.
SMALL AND STRONG, WEAK AND LARGE
 The real number system could be thought of as providing a
small model of calculus – informally, the smallest set of
numbers you could do proper calculus with.
 This is a consequence of the way real numbers are constructed –
there’s exactly one real number to act as the limit of every Cauchy
sequence because that’s what they are!
 It’s paired with a strong language that can express and prove as
many statements as possible about it.
 You can think of the hyperreal numbers as a much bigger
model – they contain the real numbers but also an infinitely
vast number more.
 They’re paired with a significantly weakened language that can’t
express certain things that seem natural – for example, although the
hyperreals include infinitesimal numbers, the logic we use can’t
describe them!
 This enables us to “dodge” contradictions that would stymie our
efforts if we used our usual, stronger logic with this bigger model.
UNIVERSES
 A universe is a special kind of construction out of sets; the
details needn’t concern us.
 Russell’s paradox teaches us we can’t have a “set of all sets”, but a
universe is supposed to be big enough to contain “all the sets we
need” for some particular purpose.
 In nonstandard analysis we actually work with two “universes”
 The standard universe, U, where ordinary analysis happens;
 The nonstandard universe, *U
 A map, written *, exists that embeds the elements of U – the
stuf f of ordinary calculus – into *U.
 Elements of *U that are in the range of this map are called standard
elements. Of course, the rest are nonstandard elements.
 Note that if a is an element of U, *a may “look” very different in *U –
it’s not “the same object” but “the corresponding object”.
LANGUAGES
 Nonstandard analysis uses ordinary classical logic, but we
don’t use it in the same way across the two universes.
 We define two languages: L, which is used to talk about U,
and *L for talking about *U.
 It’s *L that’s carefully restricted in order to avoid contradictions.
 The map * induces a new map that embeds L into *L.
 The transfer principle says that some sentence s of L is true in
U if and only if *s (a sentence of *L) is true in *U.
 All our maps embed standard things into nonstandard ones, but the
transfer principle allows us to go from a nonstandard theorem back
to a standard one, as long as we can express the theorem in L in the
first place.
SYNTHETIC
DIFFERENTIAL
GEOMETRY
MOTIVATION: PHYSICS
 Although rather fearsomely mathematical, SDG is motivated
above all by physics, from its beginnings in the work of
William Lawvere to present research in QFT and string theory.
 The quest is for a conceptually simple, elegant and unified
version of calculus that can cope with all the strange
geometric objects that crop up in contemporary physics.
 As we’ve seen on this course, calculus has a long and messy history.
It has evolved according to many different pressures and the result is
rather sprawling, messy and difficult to learn.
 SDG is an attempt to “start again”, taking what we’ve learned are the
fundamental structures and forms of thought that belong to calculus
and rebuilding them from the ground up, this time in a much more
general setting.
 This work is an ongoing research project; there are still very few
textbooks on this subject and much of the physics it’s supporting is
itself highly speculative.
QUICK HISTORY
 SDG had its beginnings in the work of Andre Weil in the
1950s; he was explicitly influenced by Fermat’s “vanishing
error” version of calculus that we saw much earlier in the
course.
 In the 1960s Alexandre Grothendieck put nilpotent elements
(which we’ll meet in a moment) at the centre of his
reformulation of algebraic geometry.
 At the same time, William Lawvere was developing topos
theory as a more general setting for calculus than the set
theory that had been used previously.
 Anders Kock wrote the first book on SDG in 1981 . There are
still very few books on the subject, and it remains a rather
specialised field in maths research.
 We’ll get some idea of why this might be, despite the stellar names
associated with it, as we go along.
INTUITION VS RIGOUR,
PRACTICE VS THEORY
L av e n d h o m me , B a s i c C o n c e p t s o f S y n t h et i c D i f f e r e n t i a l G e o m et r y
NILPOTENCY
 If a number is ver y small, then its square is ver y ver y small.

1 2
2
=
1
,
4
1 2
4
=
1
,
16
1 2
16
=
1
,
256
2
1
1000
=
1
1000000
 In real-life mathematics we often have a “ cutoff” of size we will
consider; anything sufficiently small, when squared, we set equal
to zero.
 This is just an approximation, but what if it were really true? What if
there existed some non-zero quantities that, when raised to some power
or other, became equal to zero?
 Such a number would be called nilpotent – when raised to a power
(“potent”), it becomes zero (“nil”)
 If the power is 2, we call it nilsquare.
 The key move in SDG is to construct a number system that
includes nilsquare numbers and use this as the basis of calculus.
 Notice that this looks similar to the approach of nonstandard analysis,
but we’ll end up with a different number system. In particular, we won’t
have any infinitely large or small numbers.
THE NUMBER SYSTEM
 We help ourselves to a fairly well -behaved number system R.
 In this system we can add and subtract as we like, and we have a
number we call 0 R that acts as the additive identity.
 We can also multiply as we like, and we have the multiplicative identity
1 R.
 We can divide by any whole number except 0 R .
 Multiplication and addition play nicely together, meaning we can
“multiply out brackets” (they have the property called distributivity).
 Then identify a subset of R called D, whose elements are all
nilsquare; that is, if d is in D, d 2 =0 R .
 If R is the rational or real numbers, D contains only one number, 0. We
hope we can choose an R so that D is more interesting!
 Note that the set R need not contain “numbers” in a traditional
sense; they contain objects that behave like numbers.
 This is why we write “0 R ” and “1 R ” instead of just “0” and “1” – as
reminders that these are just names for abstract objects that behave a
certain way.
R COMPARED WITH ℂ
 This quote is from the philosopher C S Peirce, cited by Bell.
 The comparison is more apt than Peirce could have known
when he made the comment.
 We can think of the complex numbers as ℝ 2 with a special
multiplication operation:
𝑎, 𝑏 𝑥, 𝑦 = (𝑎𝑥 − 𝑏𝑦, 𝑏𝑥 + 𝑎𝑦)
 We can do the same with R, but the multiplication is now
𝑎, 𝑏 𝑥, 𝑦 = (𝑎𝑥, 𝑏𝑥 + 𝑎𝑦)
 Notice the only difference is the missing “— by” in the first term.
 Intuitively, instead of i 2 =-1 we have i 2 =0.
 There’s nothing logically dodgy about this; it’s a perfectly
well-behaved algebraic object.
THE KOCK-LAWVERE AXIOM
 We now turn our attention to maps f:D R
 That is, we restrict our maps to the nilsquare elements in D, and look at
all possible ways to assign an element of R (which may or may not be
also in D) to each of them.
 The Kock-Lawvere axiom says that, for any given f, there is a
unique b in R such that f(d) = f(0) + bd
 What this says is that whatever the rule for f is, its graph is
indistinguishable from a straight line that has gradient b.
 To put it more “synthetically”, the Kock-Lawvere Axiom declares
that any the graph of function defined on the nilsquare objects D
is:
 “small enough” to be indistinguishable from a straight line ;
 “big enough” to have a direction (i.e. it’s not just a point);
 “too small” to be curved.
 In other words, if we restrict our functions to D, every function is
a straight line! That’s the connection with differential calculus.
THE KOCK-LAWVERE AXIOM
(Image from Kosecki)
 The map g:RR is not linear,
but according to the KockLawvere Axiom if we restrict
it to D it must be equal to
g(0) + bd for some constant
value b.
 Intuitively speaking, on the
very tiny region belonging to
D, the non-linear function g
is actually identical to its
linear approximation.
 A setting in which the KockLawvere Axiom holds is
sometimes called a “smooth
world”. In such worlds every
map is infinitely
dif ferentiable.
VERY CONVENIENT! BUT…
 The problem with the Kock-Lawvere Axiom is that it can easily be
used to prove that the number system R contains only one
element, 0 R . In other words, R is “trivial”. We can’t get much
geometry done in a space with only one point!
 This looks like a fatal blow for SDG.
 The proof of this fact uses the Law of the Excluded Middle – a
principle of classical logic.
 So we have an axiom that looks like it might produce an interesting
theory, and a logic that’s in contradiction with it.
 We have to throw one out – but which?
 The classical answer is to throw out the theory, of course.
 The SDG answer is to throw out the logic and replace it with a weakened
version in which the proof that R is trivial can’t be carried out.
 This should make you uneasy.
 We can prove that, if the Kock-Lawvere Axiom is true then the only
number system that could serve as R is the trivial one.
 How can weakening our logic (so we can prove less) make a suitable
number system spring into existence? Are smooth worlds the product of
mere wishful thinking, like the Big Rock Candy Mountain?
 This should remind you of the move made to define the hyperreals.
EVERY THING IS DIFFERENTIABLE
 According to the Kock-Lawvere Axiom, every function has a
linear approximation at 0 R with which it’s identical on the set
D of infinitesimals.
 Note that the “at 0 R ” part is not really a limitation, since any
function’s graph can be “moved” so that a chosen point is at 0 R .
 So we can generalize: f(a + d) = f(a) + bd
 Here the gradient b depends on a (the point at which we make the
linear approximation) but not on d (which ranges over all the
nilsquare elements in D).
 We define this linear approximation to be the derivative of the
function. That is,
 f:DR
f(d) = f(0) + bd
f’(d) = b.
 Note that we made no restrictions on f; so by our definition
every function is infinitely dif ferentiable!
 Is this a bug or a feature?
CHOICE OF AXIOMS MATTERS
 How did this happen?
 We just defined differentiation to always work through the Kock-Lawvere
Axiom.
 What’s more, there’s another axiom in SDG that defines integration to
always work, too!
 Aren’t these things we ought to prove? In fact, aren’t they false? What
about Weierstrass functions and so on?
 Functions like those can’t be defined in intuitionistic logic, so we’re
“protected” from them.
 Instead of assuming the nature of the continuum (i.e. defining
the reals in terms of limits of Cauchy sequences) and proving the
basic facts about calculus, we’ve assumed basic facts about
calculus and can now try to prove things that must be true
whenever calculus works.
 This leads to some serious philosophical questions.
 What sort of thing is legitimate to assume as an axiom?
 Is it OK to build a theory that’s trivial under classical logic simply by
weakening the logic? Are we really “protected”?
A QUICK COMPARISON
Nonstandard Analysis
SDG
 Nonsmooth maps exist
 Classical logic
 Lives in ZFC
 Equivalent to classical
analysis
 Derivatives only
approximate curves
 Infinitesimal numbers
 Only smooth maps exist
 Intuitionistic logic
 Lives in any topos
 Different theorems
from classical analysis
 All curves are “locally
straight lines”.
 Infinitesimal geometric
objects
Adapted from Bell, p.112
BIBLIOGRAPHY
 Bell, J. L. (2008) A Primer of Infinitesimal Analysis . Cambridge:
Cambridge Univer sity Press.
 The most accessible book on SDG.
 Davis , M. (1977) Applied Nonstandard Analysis . New York: John Wiley
& Sons.
 The most accessible book on the topic.
 Goldblatt , R. (1998) Lectures on the Hyperreals. New York: Springer.
 Keisler, H. J. (2000) Elementar y Calculus: An Infinitesimal Approach .
 http://www.math.wisc.edu/~keisler/calc.html
 Kock , A . (1981) Synthetic Dif ferential Geometr y . Cambridge:
Cambridge Univer sity Press.
 Kostecki, R. P. (2009) Dif ferential Geometr y in Toposes.
 http://www.fuw.edu.pl/~kostecki/sdg.pdf
 Lavendhomme, R. (2996) Basic Concepts of Synthetic Dif ferential
Geometr y. Dordrecht: Springer.
 Moerdijk , I. & Reyes, G. E. (1991) Models for Smooth Infinitesimal
Analysis. New York: Springer.
 Robinson, A . (1966) Non-Standard Analysis . Amsterdam: Nor th -Holland
Publishing Company.
 Shulman, M. ( n.d.) Synthetic Dif ferential Geometr y .
 http://home.sandiego.edu/~shulman/papers/sdg-pizza-seminar.pdf