Michael Sands
February 14, 2012

Limits in Less is Different 1

Jeremy Butterfield’s lengthy paper Less is Different undertakes a defense of awide range of theses, including claims about reduction, emergence, and supervenience.
I want to examine one strand of thought in the paper in which Butterfield offers ajustification of the use of limits in scientific idealization. This defense is based onwhat he calls the ‘Straightforward Justification’:
Straightforward Justification The use of the infinite limit is justified, despite N
being actually finite, by its being mathematically convenient and empiricallycorrect (up to the required accuracy).
These two notions of mathematical convenience and empirical correctness require
definition. The use of an infinite limit is mathematically convenient when calculations
are more easily performed after taking a limit of some parameter N as N −→ ∞. Inthese sorts of cases, calculating some value of the system for very large, yet finite, N
can be an insurmountable difficulty. This calculation may even have a computational
complexity that makes it impossible. However, treating N as if it were infinite bymeans of a limit can make the problem tractable. The use of a limit is empiricallycorrect when the error term introduced by the use of the limit can be made as small
as desired. More formally, for any ǫ > 0, with ǫ as small as one likes, the error termarising from the use of the limit can be made to be less than ǫ. Most frequently,
this ǫ will be determined by the extent of observation error induced by limitations of
instrumental measures.
1

I am grateful to Giovanni Valente for extensive comments and discussions.

1

I want to raise three issues with Butterfield’s project of justifying the use of limits.
First, his definition of a limit of a sequence of physical systems is not, in any way,a mathematical limit. Mathematical limits are defined according to strict definitions
that require the limit-taking procedure to be conducted on numerical values only.
Because systems are not composed of numerical values, neither can limits be taken ofthem, nor can appeals to mathematical convenience or empirical accuracy be made.
This criticism, if convincing, is particularly problematic for Butterfield because thesubsequent edifice of his justificatory structure is defined out of sequences of systems
and limits thereof.
The second issue I wish to raise examines a distinction he makes between two
types of quantities (observables). Essentially, Butterfield adopts a distinction that isperfectly cogent for limits of sequences of numerical values and attempts to apply it
to limits of sequences of quantities. My criticism will highlight the importance thatsequences of states of systems plays in trying to make sense of a limit of a sequenceof quantities, despite the minimal discussion given to the issue by Butterfield. I thenconclude with reasons to doubt that definitions of sequences of states of systems areforthcoming easily.
Finally, I focus on Butterfield’s attempt to apply the Straightforward Justification in the “Mysterious” case of singular limits, which are (roughly) limits where theinfinite limit provides an accurate description of a resolutely finite system. This isof particular importance to Butterfield’s thought because he seeks to respond to increased attention paid to singular limits in the literature by demonstrating that there
is nothing particularly special at singular limits.2 I raise objections to Butterfield’s
toy example, which is meant to be a paradigm of the Straightforward Justificationof singular limits. By doing so, I hope to show that the formal definition of singular2

This motivation is particularly evident in his strong disagreement with Batterman [1, 2, 3, 4]
and Rueger [8, 9]. In so demonstrating that nothing much is special about singular limits, he isroughly aligned with Belot [5] and Hooker [7].

2differs from the way Butterfield actually uses the term. His actual use relies heavily
on his definition of limits of sequences of systems and his distinction between two
types of quantities—the very things objected to in the previous two sections.
§1 will discuss the strict definition of mathematical limits for functions and sequences and draw three morals. §2 will introduce the formal terminology of Butterfield’s approach and object that a limit of a sequence of states is ill defined. §3will introduce the distinction between the two types of values for quantities andhighlight its dependence on sequences of states of systems, which I will cast doubt
upon. Finally, §4 will introduce the Mystery of Singular Limits and raise objections
to Butterfield’s toy example.

1

Mathematical Limits

There are two types of limits used in mathematics: limits of functions and limits ofsequences.

1.1

Limits of Functions

Limits of functions provide a formal statement of the informal notion that the valueof a function can be made arbitrarily close to some number, given that the input
variable is made “close enough” (but not equal) to some other value. That is,lim f (x) = L

x−→c

(1)

can be read informally as saying that the value of the function f (x) can be madearbitrarily close to L by making x sufficiently close to c. Now, the terms “arbitrarilyclose” and “sufficiently close” are imprecise and informal. They are rigorized by the
introduction of the δ − ǫ definition of limits.
3

Definition 1. Let f be a function that is defined on an open interval that containsc, except maybe at c. Let L be a real number. The equation

lim f (x) = L

x−→cmeans that for every ǫ > 0, there is some δ > 0 such that

0 < |c − x| < δ =⇒ |L − f (x)| < ǫ.

This definition can also be expressed purely symbolically as follows. For domain D:(∀ǫ > 0)(∃δ > 0)(∀x ∈ D) ((0 < |c − x| < δ) =⇒ (|L − f (x)| < ǫ)) .
Note first that limits can be taken even if the function is not defined at c. For instance,
the function

g(x) =

x2 + 1x−1

is not defined at x = 1, but the limit as x approaches 1 is defined. It is limx−→1 g(x) =2.3
Secondly, note that we can also take a limit if c = ∞. Instead of being defined
as within δ of c, a limit as x tends towards infinity is defined in terms of being bigger
than some δ. So,

lim f (x) = L

x−→∞means that for every ǫ > 0, there exists a δ > 0 such that3

A function is said to be continuous at a point when its limit and its value at that point agree.
g(x) is discontinuous at 1 because limx−→1 g(x) = 2 but g(1) does not exist. The function is,
however, continuous at all other points.

4

(x > δ) =⇒ |f (x) − L| < ǫ.
Again, a purely symbolic representation is:(∀ǫ > 0)(∃δ > 0)(∀x ∈ D) (x > δ =⇒ |L − f (x)| < ǫ) .
Third, and finally, note that a limit diverges when L = ∞. This means that for
any n ∈ R (no matter how big), if we make x sufficiently close to c, the value of thefunction at x will be greater than n. Symbolically,

lim f (x) = ∞

x−→c

means

(∀n ∈ R)(∃δ > 0) ((|x − c| < δ)Longrightarrow(f (x) > n)) .
To say a limit diverges is to say that the function can be made arbitrarily large bytaking the value of a function for x sufficiently close to c.

1.2

Limits on Sequences

Limits can also be taken of sequences. Consider a sequence of real numbers, withn ∈ N:

a1 , a2 , . . . , an , . . .
Taking a limit of this sequence consists in letting n grow without bound towards
infinity. Thus, a limit of a sequence is defined.
5lim an = L

n−→∞means that for any ǫ > 0, there exists some N ∈ N such that(n > N) =⇒ (|an − L| < ǫ) .
The relationship between limits of functions and limits on sequences should be apparent. Like limits of functions, limits of sequences can fail to exist. Additionally, alimit on sequences is said to diverge when L = ∞.

1.3

Three Morals

There are three important morals about limits to bear in mind. The first is that bothof these types of limits deal with numerical values. The reason for this restriction isthat the distance metric (in these cases the absolute value function) has to be definedsuch that the notions of ǫ and δ closeness make sense. If there is a discrete metric
under which the choice of δ is restricted, then there are possible situations where an
appropriate δ cannot be chosen.
For functions, the definition of the limit is based on choosing δ “close enough”
to c for ǫ close to L. Thus, to guarantee that for a given ǫ the range of the function
has values within the ǫ-ball around L, the domain of the function must be sufficiently
dense to ensure that there are xs within the δ-ball around c. If the domain of thefunction is not sufficiently dense, then there could be a δ-ball around c that contains
no xs for which the function is defined.
For sequences, the situation is slightly different. When taking the limit of asequence, the ns can only take on natural number values; there are no δ-balls to
be concerned with. There is, therefore, no restriction on the density of the values
6taken on by the ai . However, what are required are well-defined notions of distancefrom (when L 6= ∞) and greater than (when L = ∞), which require a metric anda well-ordering, respectively. Although it is possible to define a perverse metric andordering, the applications under consideration mean that we need only consider real
numbers. Thus, the first moral to take from the definition of limits is that limits
deal with numbers.
The second moral to take away from the definition of limits is that there is adistinction between the value at-the-limit and the value of -the-limit for limits offunctions. The value of-the-limit is L in what was defined above. However, the value
at-the-limit is the value of the function obtained by letting x = c, that is, the value
at-the-limit is f (c).
These two values—L and f (c)—need not 1) be defined, nor 2) agree with each2+1
. g(1) is undefined, butother. We saw an example of the first with g(x) = xx−1

limx−→1 g(x) = 2. An example of the second can be created by defining a piecewisefunction g1 (x) defined as follows:

g1 (x) :=

g(x); if x 6= 1
0; if x = 1

For g1 (x), the value of-the-limit and at-the-limit both exist, but they disagree (2 6= 0).
The second moral is that for limits on functions, there is a distinction between
at-the-limit values and of-the-limit values whenever c ∈ R.
The third moral applies to limits on sequences. For sequences, a limit can only
be taken when n −→ ∞ (the limit as x −→ N does not exist because the limits fromthe left and right are aN −1 and aN +1 , respectively, and they do not agree). Becauseof this restriction, whether there is a difference between the value of-the-limit andthe value at-the-limit for sequences is in one sense trivial, and in another sense, quite7difficult.
It is trivial because the value at-the-limit is not well-defined. Recall that n, thesubscript of the sequence, was a natural number. ∞ is not a natural number, so,strictly speaking, a∞ is not well-defined and does not make sense. Whether there is adifference between the value of-the-limit and the value at-the-limit is trivial because
the second is not defined, so there can be no “difference”.
The question is tricky, however, because there can be natural extensions of thesequence to ∞ and beyond. If we consider the sequence defined by an := n2 , thenit is possible to extend the sequence for n = ∞ = ω. The question whether thevalue of-the-limit equals at-the-limit can then be meaningfully asked, but only onthat extension because the calculation of a∞ is defined for that extension. The third
moral is that the at-the-limit/of-the-limit distinction for sequences is only
possible when a∞ is independently well-defined. By ‘independently’, I meanthat for the distinction to be possible in principle, the definition of a∞ cannot bemere shorthand for limn−→∞ an .
Each of these three morals will play an important role in the following criticisms.
The first will be featured in §2, while the second and third will be involved in §3 and§4.

2

When a Limit is not a Limit

2.1

Systems, Quantities, Values

In his paper, Butterfield focuses on three elements: systems, quantities, and values.
System A system is some physical system of interest denoted by σ. These systems
can take many forms. In statistical mechanics, a physical system can be somethingas simple as a cup of coffee or a box of gas composed of some number of particles.
8In probability theory—an example given by Butterfield himself—a system can be aroulette wheel divided into alternating red and black wedges.
These σ-systems are defined in terms of a parameter N ∈ N, which determinessome feature of the system, e.g., the number of particles of the gas in the box. Strictly
speaking, ‘σ’ on its own is bad notation because it denotes a class of systems rather
than an individual system. σ indicates the type of physical system we are talking
about—e.g., coffee cup or box of gas—while the choice of N selects a particular systemof that type.

Quantity Each type of system has observable quantities by which we describe features of that type of system. Different types of physical systems have different quantities. Examples of quantities include, but are not limited to, momentum, position,
temperature, energy, or spin. In the case of gas in a box, the relevant quantities might
include momentum or position.
Importantly for Butterfield’s conception of quantity, quantities are only definedover types of systems and not independently of those particular systems. Each quantity is limited to a particular type of system. Thus, a quantity f is defined over asystem σ(N) as f (σ(N)).

Value Finally, a quantity of a system only obtains a numerical value when thephysical system takes on some state. A state of a system is some arrangement of thephysical stuff that comprises the system; the state is what determines the numericalvalues of the quantities on that system. It is not until the state of the system, denotedsN , is specified that the value of the quantities can be determined. Additionally, thestate of the system characterizes the system. This dependence on states marks animportant distinction between quantities and values: quantities make no reference tostates while values depend on states. This point will become particularly important
9

later when trying to define a limit for sequences of values.
We can only talk about a numerical value when a i) particular physical system,ii) quantity, and iii) state have been specified. Accordingly, we can think of a quality
defined on a system as a function from states to numerical values, each of which
depends on the type of system (σ) and the parameter N. Notationally, a value isrepresented as v (f (σ(N)), sN ). However, Butterfield suppresses the involvement ofstates in his notation and writes values as v(f (σ(N))).
For the reader’s convenience, the notation used by Butterfield for systems, quantities, and values is as follows.
• σ(N) is the system defined by some parameter N.
• f (N) := f (σ(N)) is some quantity on the system σ(N).
• v(f (N)) := v(f (σ(N))) is the value of some quantity on the system σ(N).

2.2

Limits of Sequences of Systems, Quantities, and Values

The definition of a system, dependent as it is on a parameter N ∈ N, suggests thenotion of a sequence of systems.

σ(1), σ(2), . . . , σ(N), . . .

(System Sequence)

Note also that this sequence of systems also gives rise to a sequence of quantities
defined over those systems:

f (1), f (2), . . . , f (N), . . .

(Quantity Sequence)

Finally, assuming that there is an associated sequence of states (sN , which is suppressed in Butterfield’s notation), there is a sequence of numerical values of quantities:10

v(f (1)), v(f (2)), . . . , v(f (N)), . . .

(Value Sequence)

These sequences suggest the use of mathematical limits. Butterfield takes there to be
three questions regarding such uses. He writes,(1) One can ask whether this sequence [of systems] has as a limit, in thesense of there being (as a mathematical entity), a natural well-defined
infinite system σ(∞).
(2) One can ask whether a sequence of quantities on successive systems,
say f (N) := f (σ(N)), has a limit, which we might denote by f (∞)(Of course, the physical idea of each member of such a sequence will
be in common, e.g. energy or momentum: but we distinguish themembers by their being quantities on different (sizes of) system.)(3) Finally, one can ask whether a sequence of real number values on
quantities on successive systems, say v(f (N)) := v(f (σ(N))), has alimit.4
These questions are about the existence of limits of sequences of systems, quantities,
and values of quantities, respectively. There are two senses in which Butterfield isasking the question. The first question is whether sense can be made of the notionof taking the limit of systems, quantities, and values of quantities, respectively. The
question is whether, for instance, a limit of a sequence of systems can be well-defined.
The second sense of the question is: given that the notion of taking the limit is welldefined for sequences of systems/quantities/values of quantities, what is that limit. In
terms of our previous notation, answering the first sense of question requires providing
a definition of a limit, while answering the second sense requires a determination ofsome system/quantity/value of a quantity. The first sense of the question is about the4[6, p. 12]. A particularly puzzling aspect of Butterfield’s characterization of the limit of asequence of systems is his declaration that the infinite system is a “mathematical entity.” Systems
are meant to be physical structures that are realized in the world, not merely mathematical models.
It is not hard to imagine his motivation, given that he wants to deny (emphatically) that infinite
systems are realized. But, if that is the case, he needs to explain how a sequence of physical systems
has a mathematical entity as its limit.

11

generic definition of limits; the second sense of the question is about some particular
limit.
In the case of a sequence of values of quantities, the answer to the first sense of thequestion is affirmative, since the standard definition of limits applies. A sequence ofvalues is merely a sequence of real numbers, and we have seen above how such a limit
is defined. For systems and quantities, however, the given mathematical definition oflimits is unacceptable because of our first moral: limits can only be taken of numericalvalues. Applying a limit to sequences of systems and values requires extending theordinary definition of limit-taking beyond its numerical basis.
Butterfield takes there to be a viable candidate for such an extension of thedefinition of limits to systems and quantities. He writes,
But we can also make sense of the first two questions. As to (1): in
both classical and quantum physics we can often define the limit of asequence σ(N). Some approaches individuate a system by its state-space,
and then use infinite cartesian or tensor products (for the classical andquantum cases respectively). Other approaches individuate a system by
its set (in fact: algebra) of quantities, and then define limit algebras. This
leads to how we make sense of (2). The algebra of quantities usually has
a mathematical structure (in particular a topology) that enables one todefine the limit of a sequence of quantities (i.e. not just, as in (1), a limitof a sequence of their values).5
This is, unfortunately, all we get regarding how to extend the definition of limit
to systems or quantities. I want to focus on how Butterfield defines the limit of asequence, leaving aside for the moment the question of how to define the limit ofa sequence of quantities. I will take up the issue of defining limits of sequences ofquantities in the next section.
Taking the case of classical physics as an example, the picture is presumably thata system is described by a 6N-dimensional Cartesian space, with the six dimensions
representing the x, y, and z coordinates of position and momentum, respectively.
5[6, p. 13].

12

This N is the parameter that we let approach ∞, and thus, the limit Cartesian space
has infinite dimension.

2.3

The Problem with the “Limit” of a Sequence of Systems

Two things about this approach to defining limits of sequences are important. First,
the sequence of systems is not a sequence of numerical values. A system is a physicalsystem such as a box of gas or a cup of coffee. From the first moral above, it is
the case that limits, as defined strictly, only deal with numerical values that can beaccommodated in an ǫ-definition. So, any attempt to take a limit of a sequence ofsystems would be a complete redefinition of the limit from the strict definition above.
Second, this redefinition defines taking the limit of the sequence σ(N) as thesystem resulting from setting the parameter N = ∞. Symbolically, this approachdefines the limit in the following way.

lim σ(N) := σ(∞).

N −→∞The at-the-limit/of-the-limit distinction for sequences of systems appears to be, in
principle, impossible because the of-the-limit system is defined as the system at-thelimit. This limit-taking is, in fact, not limiting-taking behavior at all because there
is no mathematical definition of the limit. Instead, “taking the limit” of a sequenceof systems is determining what the system would look like if N = ∞.
This determination of the system when a parameter is infinite runs afoul of thesame problem that prompted our third moral above: taking the limit only makes sensewhen the case where N = ∞ is well-defined. However, one of the main applications oflimit taking is for systems where physical systems with N = ∞ is not a well-definedsystem. Given that the cases under examination are precisely those where N = ∞is not a well-defined notion for the physical systems (e.g., the number of particles in13

a system cannot be infinite), defining the limit of a sequence of systems as equal to
the system at-the-limit presents the first major problem with Butterfield’s analysis.
He sets out to justify the use of limits in scientific idealizations, but by defining the“taking of a limit” of a sequence of systems as considering N = ∞, there is no moretaking of limits to be justified for systems.

3

Limits of Sequences of Quantities

Recall that Butterfield’s project seeks to justify the use of infinite limits in scientificidealization using the straightforward justification of mathematical convenience andempirical accuracy. However, in order to apply this justification to all instances ofmathematical limit taking, Butterfield must respect a distinction I drew earlier for
limits on functions between values of-the-limit and values at-the-limit. This distinction allows Butterfield to address the “mystery” of singular limits, which are definedin the following way.
Singular Limit: An infinite limit is singular when
1. The value of-the-limit exists;
2. The value at-the-limit exists;
3. They do not agree;
4. The value at-the-limit is empirically accurate.
In the current section, I will to detail exactly how Butterfield conceives the at-thelimit/of-the-limit distinction and argue his application of the distinction to limits ofsequences of quantities depends crucially on a notion that he suppresses: sequencesof states of systems. I will close the section with reasons to doubt that sequences ofstates are simple enough to be suppressed. §3.1 is of continued importance in §4, for
it is through this distinction that singular limits are defined.
14

3.1

Drawing the Distinction

Butterfield draws the of-the-limit/at-the-limit distinction for both quantities and values. He starts with two suppositions. Supposea) a sequence v(f (N)) of values of a quantity has a limit limN −→∞ v(f (N))
as N tends to infinity (as mentioned in Section 3.1, a sequence of statessN is here understood, so that one might write v(f (N), sN )). Andb) there is also a well defined infinite system σ(∞) on which: b.1)] thecommon physical idea of the various f (N) makes sense and gives a natural
well- defined limit quantity, which we might write as f (σ(∞)) (on σ(∞));
and b.2)] on which there is a natural well-defined limit state, s say.
Then we need to distinguish:(i) the given limit of the values, limN −→∞ v(f (N)) ≡ limN −→∞ v(f (N, sN ))from(ii) the value v(f (σ(∞), s)) of the natural limit quantity f (σ(∞)) in thenatural limit state, s.6

In the quotation, (i) is the value of-the-limit and (ii) is the value at-the-limit. To
reiterate from the earlier discussion of limits on sequences, the of-the-limit value isthe result of applying a calculus limit to a sequence of values changing across a single
parameter. The at-the-limit value is the result of substituting ∞ for the parameterand calculating the resultant value. The at-the-limit value only makes sense giventhat there “is a well-defined infinite system σ(∞).” The at-the-limit value of thesequence is the (ω + 1)th element in the sequence since we are taking the limit as
N −→ ∞ = ω.
The distinction can be illustrated most clearly by considering a sequence of sets(systems) that contain the natural numbers:
6[6, p. 14].

15

σ(1) = {1}σ(2) = {1, 2}..
.

Let f (σ(N)) be the ordinal type function, ord(σ(N)). The values of the function
are as follows:v(f (σ(1))) := v(f (1)) = 1v(f (σ(2))) := v(f (2)) = 2..
.
v(f (σ(i))) := v(f (i)) = i..
.

This sequence of values increases as N increases. We can also take a limit of thissequence of values,

lim v(f (N)) = ωN −→∞This limit has the form of the ordinary calculus limit—it is the result of determining
the value of a numerical quantity obtained by letting a single parameter get increasingly large towards ∞. The limit diverges because for any M ∈ N, there exists
an N ∈ N such that v(f (N)) > M. Note that for our example, the limit is equal
to the ordinality of the natural numbers, which makes sense given that taking thelimit in some way “runs through” all of the natural numbers and “arrives” at the set
containing all of the natural numbers.
16However, the value of-the-limit is not the same as the value at-the-limit. In thiscase, the at-the-limit system is obtained by letting N = ω. But, v(f (ω)) 6= ω becauseσ(ω) does not have order ω. Recall that for every i,

σ(i) := {1, 2, . . . , i}Thus,
σ(ω) := {1, 2, . . . , ω}which has order ω + 1. Thus, v(f (ω)) = ω + 1. In this example, there is a numericaldifference between the value of-the-limit (ω) and the value at-the-limit (ω + 1). Thisdifference is only possible when N ∈ N ∪ {ω}.

3.2

Distinction for Quantities

Butterfield wants to draw this same distinction for quantities that he does for values.
In fact, he takes it that the distinction is identical, save for the involvement of statesof the system. He writes,
For quantities themselves, rather than values, the point is in essence thesame. The statement is a close parallel of that in (a): indeed, shorter since
we refer only to quantities, not to values of quantities—albeit thereby
more abstract. Thus suppose: (i) a sequence of quantities f (N) has alimit, dubbed f (∞) in Section 3.1. And suppose also: (ii) there is alsoa well-defined infinite system σ(∞) on which the common physical ideaof the various f (N) makes sense and gives a natural well-defined limit
quantity, which we might write as f (σ(∞)) (on σ(∞)). Then we need to
distinguish:(i) the given limit, f (∞) := limN −→∞ f (N), from(ii) the natural definition of the quantity f (σ(∞)) on σ(∞).7
The distinction that Butterfield means to draw, then, is between the quantity on theinfinite system and the limit of the sequence of quantities. Because of the definition
7[6, p. 12].

17

of f (N) and the fact that σ(∞) = limN −→∞ σ(N)8 , the distinction drawn is between
limN −→∞ f (σ(N)) and f (limN −→∞ σ(N)). The claim is that in principle, these twoquantities can be different. They need not be different, but they can be.
Given the suppositions of the distinction as laid out, the definition of the quantity
on the infinite system is well-defined. Therefore, f (limN −→∞ σ(N)) is a well-definedquantity. The question then becomes how to make sense of limN −→∞ f (σ(N)). Recall
first the strict definition of limits, which deal only with numerical values. For a sequence to have a limit L, for any arbitrary ǫ > 0, there must be some element in thesequence for which the distance between L and any later element of the sequence is lessthan ǫ. However, in the case of sequences of quantities, no such convergence (or divergence) is possible, since quantities are not numerical values. The limN −→∞ f (σ(N))
cannot be defined in terms of limits on numerical sequences.
It might, however, be defined as a limit of a sequence of functions. Pointwise
convergence of a sequence of functions is given by the following general definition.
Given a domain D,

lim fN (x) = g(x) ⇐⇒ (∀x ∈ D)

N −→∞



lim fN (x) = g(x)

N −→∞



In words, this definition means that a sequence of functions fN approaches (convergesto) a function g(x) if for every x ∈ D, every sequence of values of fN (x) converges tothe value of g(x). In other words, to prove pointwise convergence of fN (x) to g(x),one must show that for some ǫ > 0 and every x, there is some m ∈ N such that|fm (x) − g(x)| < ǫ for every N > m.
There are a few important things to take from the definition of pointwise convergence. First, it is built up out of limits of sequences of values; the left hand side ofthe definition refers only to limits of functions, while the right hand side only refers
8

By definition, see p. 13.

18

to limits of sequences of values of functions. Second, note that the domain of thefunctions fN (x) and g(x) is identical and fixed. Both fN (x) and g(x) go from D tonumerical values.
Therefore, in order to apply the pointwise definition of limits to quantities, wemust think of quantities as functions. But, in order to treat them as such, we mustfirst determine their domain and range. The latter is the easier of the two, for it ismerely numerical values, be they real, natural, rational, or what have you, dependingon circumstance and the physical interpretation of the quantity. In general, thedomain of the quantities when considered as functions consists of states of systems.
It is also important to clarify the role of the parameter N in the reimaginingof quantities as functions. N determines the dimension of the states of the system,and therefore, N determines the dimension of the state space. In this way, N also
determines the shape of the functions; a state with N-many dimensions will give rise
to a function that takes N-many inputs.
With these clarifications in hand, we can address how Butterfield seeks to apply
pointwise convergence of sequence of functions to sequences of quantities. Unfortunately, the notation used by Butterfield obscures the point I am trying to make, so
for the purposes of this definition, I will take quantities in the following way.

f (σ(N))(x̄) : SN 7−→ R
is a function defined for a system σ(N) that goes from the state space of the system(SN ) to R We can then apply pointwise convergence in the following way, bearing in
mind that x̄ in this context refers to a state of the system, which can be of any finite
dimension. Given SN , which is the state space with dimension determined by N,

19

lim f (σ(N))(x̄) = g(σ(∞))(x̄) ⇐⇒(∀x̄ ∈ S∞ ) lim v(f (σ(N))(x̄) = v(g(σ(∞)))(x̄)

N −→∞

N −→∞

This definition of pointwise convergence of one quantity to another quantity is adirect analogue of the definition of pointwise convergence above. Again, the left handside of the definition deals entirely with quantities (functions), while the right handside of the definition only refers to values of quantities (functions). Additionally, thedefinition, as before, quantifies over all elements in the domain on the right handside. So, the definition of pointwise convergence of quantities is built up out of theconvergence of values of that quantity for every state in the state space.
However, in defining convergence of quantities in this way, a problem arises. In
the standard definition of pointwise convergence, the domain is fixed. However, inthe definition of pointwise convergence for quantities, two things are converging: thevalues of the functions and the domains of the functions. Each quantity f (σ(N))(x̄)
is only defined over state spaces that have N-many dimensions. That is, f (σ(N)(x̄)
is a function from state spaces of dimension N to the real numbers; it is, for instance,
not defined over state spaces that have infinitely many dimensions, i.e. for S∞ . So, inorder to have a change at being well-defined, there must be a way to make the sequenceof states of the system “line-up” appropriately with the sequence of quantities that
we are taking the limit of.
Unfortunately, the notion of taking limits of states of systems is treated exceedingly briefly by Butterfield. The entirety of his comments on the matter is as follows:[Regarding limits of values] Here a sequence of states, sN say, on the σ(N)is to be implicitly understood, so as to define values for the quantities
f (N); but to simplify notation, I will for the most part not mention sN ,20and indeed take states as understood.9

In the definition of a sequence of values, the states can be suppressed, since they are
input variables and all we care about in that context is the output values. However,the definition of a sequence of quantities must quantify over all possible states in theinfinite state space, so Butterfield cannot rely on a single fixed sequence of states.
There must be some account of taking limits of sequences of states of systems.
As an example of the complexity involved in taking limits of sequences of states,consider a greatly simplified example. Let σ(N) be a system with N elements that
can take the value of 1 or 0. Let s∞ be a state of this infinite system given by thefollowing sequence of values for elements: {α1 = 1, α2 = 0, α3 = 1, α4 = 0, . . .}.
Suppose further that the value under consideration is the numerical average. of all
the terms in the sequence. Then, there are two different sequences of states thatconverge to a∞ (in one sense), but give vastly different values. Consider the following
two sequences:

Sequence 1 :{1}, {1, 0}.{1, 0, 1}, {1, 0, 1, 0}, {1, 0, 1, 0, 1}, {1, 0, 1, 0, 1, 0} . . .
Sequence 2 :{1}, {1, 1}.{1, 1, 0}, {1, 1, 0, 1}, {1, 1, 0, 1, 1}, {1, 1, 0, 1, 1, 0} . . .

These two sequences both “converge” to a∞ in the sense that both sequences canhave their 1s and 0s put into one-to-one correspondence with the respective 1s and 0s
in a∞ . However, if we take the limit of the numerical average of the terms containedin Sequence 1 and Sequence 2, we find that the limit of the averages are 21 and 32 ,respectively. So in one sense of “convergence”, these two sequences are the same, but
their values differ for a choice of quantities.
9[6, p. 12].

21

Drawing the distinction between of-the-limit quantities and at-the-limit quantities, then, rests crucially on assumptions that underlie taking the limit of a sequenceof states, at least if we are to define it in terms of a limit of a sequence of functions,which seems to be the only possible definition. This underlying framework is absent
from Butterfield’s account. For the time being, I will put to the side the issues underlying limits on quantities and focus solely on limits of values, a concept that, forthe moment, does not have any problems.

4

Singular Limits and Toy Examples

4.1

The Mystery of Singular Limits

Recall that Butterfield seeks to apply the Straightforward Justification of mathematical convenience and empirical accuracy to the use of mathematical limits in scientificidealizations. The distinction between values of-the-limit and values at-the-limit gives
rise to a four-part classification of limits.

(I) The value of-the-limit exists and the value at-the-limit does not exist;(II) The value of-the-limit does not exist and the value at-the-limit exists;(III) The value of-the-limit exists and the value at-the-limit exists and they agree;(IV) The value of-the-limit exists and the value at-the-limit exists and they disagree.
The appeal to the Straightforward Justification in Cases I-III is immediately
expected, according to Butterfield. For cases I and II, there is only one value toconsider, so that the value is empirically accurate is sufficient to justify the use of
the limit. Similarly, case III presents only one value because the at-the-limit andof-the-limit values are equal.
22

Only Case IV presents a problem, and even then according to Butterfield, only
when the value at-the-limit is the empirically accurate value. The reason that thisconstitutes a mystery is that the systems for which the values of the quantities aredefined are strictly finite systems. If the value at-the-limit—the value of the quantity
as defined on the infinite system σ(∞)—is empirically accurate despite the existenceof a value of-the-limit, then that accuracy seems to imply that the physical system
has infinitely many degrees of freedom. Returning again to the box with N-manygas molecules, a singular limit would imply that the physical system has an infinite
number of molecules, a result that directly contradicts the finitude of matter. Indeed,
all the physical systems under consideration have only a finite number of degrees of
freedom. Thus, the Mystery: how can Butterfield Straightforwardly Justify empirically accurate uses of values at-the-limit when their physical interpretation violates
basic physical principles?

4.2

Dissolving the Mystery of Singular Limits

The mystery of singular limits, according to Butterfield, can be dissolved by appealing
to values of other functions, which achieve empirical accuracy long before the infinite
limit. The mathematical convenience of the Straightforward Justification attaches tothe infinite limit, while the empirical accuracy attaches to both the infinite limit andthis separate, finite cousin-function. The calculation using the value at-the-limit ismerely a convenient way of approximately (within ǫ) the actual value of the finite
cousin function.
The picture is clarified through symbolic representation. Let f (σ(N)) be somequantity on a physical system σ(N), and let the limit of the sequence of the values
be singular. That is, let23

lim v(f (σ(N))) 6= v(f (σ(∞))),N −→∞and let v(f (σ(∞))) be empirically accurate. Butterfield’s claim then is that for every such quantity f (N), there is some other quantity g(N) such that for some N,v(g(σ(N))) is an empirically accurate value for f (σ(N)). Butterfield writes,[In singular limits,] there are other quantities, for which (despite f ’s singular limit) the finite-N model, for large N, is close to the values given by
the infinite model and is thereby also empirically correct. In fact, theseother quantities are ‘cousins’ of the quantity f which we first considered.10
He further illustrates this method of cousin functions using a toy mathematical example. Consider a sequence of real functions:



−1 :
x ≤ − N1gN (x) =Nx : − N1 ≤ x ≤ N1




1
 1:
≤x
N
Butterfield takes it that “the sequence has as its limit the function g∞ given by:”11 − 1:x<0
g∞ (x) =
0:x=0





1:x>0
The sequence of gN s is meant to exemplify the mystery of singular limits along thefollowing lines. Every gN with finite N is a continuous function, but g∞ is discontinuous, making the limit ‘singular’ because “continuity is lost.”12 Further, supposethat calculating gN for very large N is difficult while using g∞ is mathematicallyconvenient. All that is left to demonstrate the mystery of singular limits is empirical
10[6, p. 16].
[6, p. 17].
12
[6, p. 17].
1124

accuracy, which Butterfield takes it, is guaranteed for large enough N. Then, if N
is large enough, the shape of the graph of values of the function will be sufficiently
close to g∞ (x) that it is empirically indistinguishable. That is, given a physical interpretation of gN (x) and g∞ (x), no measurement could differentiate between gN (x)and g∞ (x).

4.3

Problems with the Toy Example

However, the problem with the above reasoning is that the limit of the sequence ofgN s is not singular by Butterfield’s own lights. Recall that part of being singular isa disagreement between the values of-the-limit and at-the-limit. However, I claimthat Butterfield has explicitly defined the limit of the sequence of gN s as g∞ . He
writes, “This sequence has, as its limit, the function g∞ , defined by . . . ”13 This“limit-taking” behavior is not defined over the whole domain of the functions because
g∞ is discontinuous at x = 0. The sequence of gN s do not converge pointwise to g∞
precisely because of that discontinuity. Put another way, consider a different infinite′′
function g∞, which is identical to g∞ except g∞(0) = 1. This new function agrees

with the limit of gN (x) to the same extent that g∞ does. There is no difference,from the point of view of taking a limit of the sequence of gN s, between g∞ and′ 14g∞. Thus, Butterfield is not taking a limit in referring to g∞ , but rather explicitly

defining limN −→∞ gN (x) = g∞ (x), which means that this limit cannot be singularbecause there is no disagreement between values of-the-limit and at-the-limit.
In order to gin up the needed disagreement, Butterfield introduces a two-valued
quantity fN such that,fN , N ∈ N ∪ {∞} that encodes whether or not gN is continuous: fN := 013[6, p. 17].
There are good practical reasons to prefer g∞ over other functions, for instance the fact thatgN (0) = 0 for all N . But this practical consideration does not make the sequence of functions
converge to g∞ in any mathematical sense.
1425

if gN is continuous and fN := 1 if gN is discontinuous. Then we have:fN = 0 for all finite N ∈ N, but f∞ = 1.”15

Because fN = 0 for all N ∈ N, limN −→∞ fN = 0. However, f∞ = 1. Taking thelimit of the fN s provides genuine disagreement between the values of-the-limit andthe values at-the-limit.
But a counterpart of the previous problem arises for the fN s when considering
whether the limit is singular or not. Although there is genuine disagreement, it is
not at all clear that the f∞ is empirically accurate while limN −→∞ fN is not. If thefunction under question is some gN with finite N, then the empirically accurate f function is fN , for every gN is continuous. It would be empirically inaccurate to use
f∞ to describe the continuity of any gN , with finite N. Indeed, the only empiricalaccuracy appealed to by Butterfield is the empirical accuracy of g∞ .
Even in this simple, toy mathematical example, a number of issues arise. First,
taking the limit of the sequence of functions does not automatically generate an infinite function for all potentially relevant values. Only by defining the infinite function
in a particular way does the infinite function gain its empirical accuracy by which
it is meant to be straightforwardly justified. But the Straightforward Justification ismeant to apply to the use of limits, not merely the definition of infinite functions, andin this case, limit taking is not taking place. Second, even though the toy example
is meant to illustrate how to resolve the mystery of singular limits, it is not at all
clear that we have a singular limit. The limit of the sequence of the gN s (apart fromx = 0) agrees with g∞ , so there is no disagreement between the value of-the-limit andthe value at-the-limit. The limit of the sequence of fN s and f∞ disagree, but it is notclear what f∞ is meant to be an empirically accurate description of.
15[6, p. 17].

26

5Conclusion

The problems of the toy example generalize. The main problem stems from a discrepancy between how Butterfield defines singular limits formally, and the way he uses
the term in his examples. His use of ‘singular’ indicates that the value at-the-limit
exhibits some feature that does not appear for any finite value. More precisely, it is
not the value that exhibits this new feature, but the infinite system or quantity. This
point can be witnessed in the toy example, for Butterfield appeals to a feature about
the quantity (function)—its continuity or lack thereof—to demonstrate that the limitis ‘singular’. Singularity, as actually used by Butterfield, is a claim about a uniquefeature of the at-the-limit quantity or system. But I hope that §2 and §3 raised issues
for Butterfield’s definitions of limits of sequences of systems and quantities.

27

References[1] Robert W. Batterman. The Devil in the Details. Oxford University Press, 2002.
[2] Robert W. Batterman. Hydrodynamic vs. molecular dynamics: Intertheory relatioons in condensed matters physics. Philsophy of Science, 73:888–904, 2006.
[3] Robert W. Batterman. Emergence, singularities, and symmetry breaking. Pittsburgh PhilSci Archive, 2009.
[4] Robert W. Batterman. The tyranny of scales. Manuscript, June 2011.
[5] Gordon Belot. Whose devil? which details? Philsophy of Science, 72:128–153,
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[6] Jeremy Butterfield. Less is different: Emergence and reduction reconciled. Foundations of Physics, 41:920–959, 2011.
[7] Cliff Hooker. Asymptotics, reduction and emergence. British Journal for thePhilsophy of Science, 55:435–479, 2004.
[8] Alexander Rueger. Physical emergence, diachronic and synchronic. Synthese,
124:297–322, 2000.
[9] Alexander Rueger. Functional reduction and emergence in the physical sciences.
Synthese, 151:335–346, 2006.

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