1. The Return of the Infinitesimal?
History & Philosophy of Calculus, Session 10

2. 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.

3. 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 difficult 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.

4. Synthetic Geometry

5. New Geometries
Analytical geometry had huge successes in the 17th and 18th 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 19th 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.

6. Hegel
In the Science of Logic , Hegel asserts that analytical geometry has taken a wrong turn.
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.

7. 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.

8. Two Different Projects
Moerdijk & Reyes, p. v

9. Nonstandard Analysis

10. 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.

11. 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.

12. 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.

13. 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 stuff 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”.

14. 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.

15. Synthetic Differential Geometry

16. 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.

17. 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 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.

18. Intuition vs Rigour, Practice vs Theory
Lavendhomme, Basic Concepts of Synthetic Differential Geometry

19. Nilpotency
If a number is very small, then its square is very very small.
= 1 4 , = 1 16 , = , =
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.

20. 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 0R that acts as the additive identity.
We can also multiply as we like, and we have the multiplicative identity 1R.
We can divide by any whole number except 0R.
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, d2=0R.
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 “0R” and “1R” instead of just “0” and “1” – as reminders that these are just names for abstract objects that behave a certain way.

21. 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 i2=-1 we have i2=0.
There’s nothing logically dodgy about this; it’s a perfectly well-behaved algebraic object.

22. 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.

23. The Kock-Lawvere Axiom
The map g:RR is not linear, but according to the Kock-Lawvere 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 Kock-Lawvere Axiom holds is sometimes called a “smooth world”. In such worlds every map is infinitely differentiable.
(Image from Kosecki)

24. 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, 0R. 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.

25. Everything is Differentiable
According to the Kock-Lawvere Axiom, every function has a linear approximation at 0R with which it’s identical on the set D of infinitesimals.
Note that the “at 0R” part is not really a limitation, since any function’s graph can be “moved” so that a chosen point is at 0R.
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 differentiable!
Is this a bug or a feature?

26. 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”?

27. 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

28. Bibliography
Bell, J. L. (2008) A Primer of Infinitesimal Analysis. Cambridge: Cambridge University 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) Elementary Calculus: An Infinitesimal Approach.
Kock, A. (1981) Synthetic Differential Geometry. Cambridge: Cambridge University Press.
Kostecki, R. P. (2009) Differential Geometry in Toposes.
Lavendhomme, R. (2996) Basic Concepts of Synthetic Differential Geometry. Dordrecht: Springer.
Moerdijk, I. & Reyes, G. E. (1991) Models for Smooth Infinitesimal Analysis. New York: Springer.
Robinson, A. (1966) Non-Standard Analysis. Amsterdam: North-Holland Publishing Company.
Shulman, M. (n.d.) Synthetic Differential Geometry.