Pebble
and Gingerbread Arithmetic Redux
In
this essay I will present an exposition of John Stuart Mill’s account of
arithmetic as found in book two (section VI) of System of Logic, followed by an analysis of Gottlob Frege’s
arguments against Mill’s position found in The
Foundations of Arithmetic. I will initially elucidate Mill’s account of
arithmetic and reveal it to consist of the propositions a)
mathematical statements are reliable by virtue of their acquisition from inner
consciousness or sense-experience that hold true of the task of revision, and
meaningful by their consequent application within a context; b) proving
mathematical statement a involves
providing the necessary empirical basis for stating a as valid; and c) mathematics is a subset of the deductive
sciences, which itself exists as a subset of the inductive sciences. I will
then provide an account of Frege’s direct responses to Mill’s account (the
three claims stated above) and show that Mill’s so-called ‘pebble and
gingerbread arithmetic’ fails in Frege’s eye to do justice to the objectivity
of number; Frege will argue that: a) Mill’s account lacks implimentability and
cohesiveness, b) Its unlikely that the laws of arithmetic are contingent of
whether certain empirical facts obtain or not, namely a fact pertaining to
one’s psychological capacity, and c) Mill disregards the logical necessity
underlying number and instead focuses on the application of said statements in
the empirical realm.
Mill
begins his discussion in section VI with a recapitulation of claims made in the
previous chapter concerning the nature of ratiocination in the demonstrative
sciences; while Mill acknowledges that ‘the results from these sciences are
indeed necessary’ (164 Mill), he claims that they are only necessary insofar as
the axioms that guarantee their certainty obtain, or are at least considered to
be undeniable by virtue of ‘superabundant or obvious evidence’. (166) This view
departs, Mill claims, from the view that axioms are somehow superior in
relation to objectivity than other instances of ratiocination; while the
indubitably of first principles might lead one to conclude they arise from ‘evidence
independent of and superior to observation and experience’ (165), Mill argues
that axioms are ‘but a universal class of inductions from experience; the
simples and easiest cases of generalisations from the facts furnished to us by
our senses or internal consciousness’. (166) The conclusion that naturally
arises from this according to Mill is that the deductive sciences are in some
sense always inductive; they are formed by virtue of an inference drawn from
evidence and are therefore always hypothetical in some fundamental sense.
However, Mill is wary to proclaim this true of all deductive sciences in one
fell swoop and therefore decides to test his claim against what he calls ‘the
most remarkable of all sciences’ (166): the science of numbers.
Mill
carries out his discussion on the science of number by first presenting two
doctrines he claims attempt to provide an accurate account of the nature of
number and arithmetic, and secondly by making his theory of the nature of number
explicit by elucidating it in relation to what he perceives as the shortcomings
of the aforementioned doctrines. The
first of these doctrines is the one held by ‘a priori’ philosophers, who Mill claims argue that the certainty
present in mathematics is due to the objective a priori existence of the essential truths of number; something
that would, as stated above, consist of ‘evidence independent of and superior
to observation and experience’. (165)[1]
The second view is one in which, according to Mill, arithmetic is seen as ‘merely
verbal, and its propositions as simple transformations of language’. (166) This
view, Mill argues, conceives of arithmetic as the transmogrification of any
statement a into a more complex set
of symbols that retain the initial meaning of a; mathematics as mere symbol manipulation.
Mill
is quick to highlight what he perceives as two shortcomings of the linguistic
account of number: a) the linguistic account does not account for ampliative
claims, and b) it does not consider the meaning that mathematical symbols stand
for. In regards to the first issue, Mill argues that the fatal problem with the
linguistic account is that it views mathematics as a simple reformulation of propositions
already known into more robust formal language. This is simply false, Mill
argues, insofar as mathematical processes can provide one with new information;
how could it simply be a transmogrification if a theorem brings about a new
product? The first objection is interwoven with the second insofar as the
notion of new information arising out of a mathematical formulae is
incompatible with that of mathematics a purely symbol oriented; is the new
symbol in itself the new information- is a symbol ever revelatory in-itself?
Mill argues that mathematics is not merely manipulating symbols, but an
inference driven activity based on evidence and axioms held to be necessary;
given certain axioms taken to be necessary and the initial conditions of a
problem, one carries ‘a real induction, a real inference of facts from facts’ and
draws the necessary conclusions. (167)
While
Mill argues against the notion of arithmetic as mere symbol manipulation, he
claims to understand why such a confusion would arise naturally out of the
seemingly obscure nature of mathematics; while propositions in geometry carry
with them intuitions insofar as they can be conceived of spatially (one can
conceive of the lines AB and AC), the notions of algebra strike many as empty
due to the abstract symbols used (a, b, c)
and the lack of content they seem to possess. Mill builds on this by arguing
that: a) numbers only have meaning in relation to their existence within a
context (no abstract numbers exist), and b) that while mathematics is concerned
with all things numerable, algebra is concerned with the form of all
mathematical statements at large; making the proposition a simply an instance of all possible number. This does not entail
meaningless symbol manipulation, but instead a meaningful ratiocination from
number that arises contextually (ten teeth, ten bodies), to the notion of
number as applying to all things (everything is numerable), to a search for
statements that are true of all number; statements like ‘equals added to equals
make equals’ and ‘equals taken from equals leave equals’. (169) It is clear
according to Mill, that these inductive truths are not true of symbols, but only
of mathematical propositions/thoughts.
Mill
continues by assessing another attraction of the idea that statements of
algebra are merely verbal; ‘when considered as propositions respecting things,
(the propositions of arithmetic and algebra) all have the appearance of being
identical propositions’. (168) Mill uses the example ‘two pebbles and one
pebble are equal to three pebbles’ to communicate and instance of this
assumption. Mill argues that while the linguistic account would characterise
these as equivalent statements, this idea does not appear to pass the test when
one considers the distinction between connotation and denotation. While the
statements refer to the same collection of objects (their denotation), they bear
a different connotation (or sense); they express different meanings insofar as
they ‘do not make the same impression on our senses’. (168) This is important
for Mill insofar as facts do not arise out of definitions, but instead
definitions arise out of an aggregate of facts. He illustrates this thought by
drawing an example from geometry; the definition of a circle as ‘a figure
bounded by a line which has all its points equally distant from a point within
it’ does not, Mill argues, create a new symbol, but instead holds true of an
object once held to be true of experience. This account is another instance of
Mill’s assumption that all arithmetical propositions have as their basis sense
experience, and even pushes Mill to state further that this fact is obvious by
virtue of its use in ‘all the improved methods of teaching arithmetic to
children’. (169)
Mill
moves on to what he claims is the assumption that is commonly held to be the
securest amongst all mathematical ‘truths’; namely that ‘all the numbers are
numbers of the same or of equal units’; simply that 1=1. (170) Mill claims even
this tenant of mathematics is built on fragile empirical grounds; for is it the
case that in actuality, ‘while it is certain that 1 is always equal in number
to 1’, that the objects that serve as the empirical foundation for these
numbers are numerically identical? (170) Mill argues that, while it may be
assumed that one bag of flour is equivalent to another, this is a hypothetical
assumption made for pragmatic reasons and thus casts a light of contingency on
what is commonly hailed as the most certain of all axioms.
Mill
claims that, insofar as he has shown these facts to obtain, arithmetic is
revealed to be no different than the demonstrative sciences insofar as arithmetic
is also inherently inductive and hypothetical; its axioms arise either out of
an aggregate of observational instances or pragmatic choices, and mathematical
propositions themselves are empirically laden insofar as number finds its basis
and validity within an empirical context. Therefore Mill’s view of mathematics
can be reduced in some sense into three basic propositions: a) mathematical
statements are reliable by virtue of their acquisition from inner consciousness
or sense-experience that hold true of the task of revision, and meaningful by
their consequent application within a context; b) proving mathematical
statement a involves providing the
necessary empirical basis for stating a as
valid; and c) mathematics is a subset of the deductive sciences, which itself
exists as a subset of the inductive sciences.
Mill
enters Frege’s The Foundations of
Arithmetic as a philosopher who is relevant in regards to the question:
‘(a)re mathematical formulae provable?’. (Frege 5) After considering the
attempts of Kant and Leibniz to provide such a proof and concluding their
answers insufficient, Frege looks to Mill’s account of the empirical basis of
proof as potentially satisfying answer to this question. Frege begins by
stating that, while Mill might be on the right track by basing the science of
number on definitions, ‘this spark of sound sense is soon extinguished thanks
to his preconception that all knowledge is empirical’. (9) In response to
Mill’s claim that the definitions of number secure the meaning of a term while
also asserting an observed matter of fact, Frege argues that some numbers would
be meaningless by virtue of this definition insofar as there is no conceivable
instance in which a fact would represent them. Frege demonstrates this argument
simply by asking: ‘what in the world can be the observed physical fact which is
asserted by the definition of the number 777,864?’ (9)
Frege
then moves on to what he claims is Mill’s proof for the number 3: ‘a collection
of objects exists, which while they impress the sense thus, ... ,
maybe be separated into two parts, thus, .. . ’. (9) While Frege jokingly
states his gratitude at the fact that all objects in the world are not nailed
down, as 2+1 would not be provable, he states further: ‘what a pity Mill did
not illustrate the physical facts underlying the numbers 0 and 1’. (9) Frege is
arguing once again that by Mill’s notion of the definition of number, certain
numbers seem to lose their meaning; the tacit premise is that it is not clear as
to how or what physical fact could provide a satisfactory empirical basis for
the mathematical propositions 0 and 1. Frege argues further that common
instances of the use 3 would not be justified by the notion of three as the
parcel: ... . For, Frege asks: is it
correct to speak of ‘three strokes when the clock strikes three’ or ‘sweet,
sour, and bitter as three sensations’ if they do not impress the senses thus: ... ? (9) While
Frege’s major issue with Mill’s account at this juncture is the claim that all
statements of number are reducible to conceiving of a way in which said number
would appear to the senses, this issue grows when Frege asks, concerning the
proposition 999,999+1, what physical fact could allow one to conclude the
product is 1,000,000? Frege argues that by Mill’s definition ‘a fact should be
observed for each number’, thus making it taxing to add 999,999 and 1 insofar
as one would first have to provide all the facts for the numbers leading up
from 1 to 999,999, and then recapitulate that exposition to show all of those
facts remain intact once adding the fact that represents 1. (10)
Frege
proceeds to tackle what he believes to be the underlying assumption guiding
Mill’s work concerning arithmetic, namely that arithmetic exists solely by
virtue of experience. Frege argues that this concern does not take into account
the truth of arithmetic and only considers the psychologistic element; Frege
finds it absurd to state that truth is contingent on whether there exist beings
such that they can observe it. For instance, Frege asks what would be the state
of arithmetic if human beings could not make distinctions between things by
virtue of the senses; is it reasonable to say the truth of arithmetic is
contingent on whether one does or does not have such a capacity? Frege states
that ‘the question of truth’ remains untouched by Mill’s empiricist notion of
arithmetic.
In
an effort to further assess Mill’s view of arithmetic, Frege moves away from
the question of whether statements of number are provable, and instead asks
more generally: ‘are the laws of arithmetic’, as Mill claims, ‘inductive
truths’? (12) Frege proceeds to take what Mill calls an inductive truth the
highest order, namely that ‘the sums of equals are equals’ and ask: what makes
such a law inductive? Frege claims that Mill fails to see the necessity of such
propositions due to the fact that he ‘attributes to them a sense which they do
not bear’; when focusing on the above mentioned assertion by Mill that 1=1
could be false if the facts that underlie 1 and 1 are not numerically
identical, Frege simply states that the ‘proposition 1=1 is not intended in the
least to state that it (would)’. (13) Frege makes a crucial point when stating
that Mill is confused by the distinction between ‘the applications that can be
made of an arithmetical proposition, which often are physical and do presuppose
certain observed facts, with the pure mathematical proposition itself’. (13)
There’s a sense in which, for Frege, the relevant concern in this instance is
one of a logical nature and not empirical applicability; in the same way that
Frege argues the meaning of ‘+’ may have applications in the physical world in
sense of representing ‘heaps or aggregates’ but ultimately retains a logical
identity in-itself, the proposition ‘1=1’ possesses a logical sense that is not
contingent on whether two bags of flour are numerically equivalent. (13) This
leads Frege to conclude that addition cannot be a law of nature insofar as its
logical identity is not determined by whether certain physical facts obtain or
not.
In
conclusion, while Frege repeatedly features Mill’s account throughout the
remainder of The Foundations of
Arithmetic, it would be fair to say that my account thus far illustrates
Frege’s direct arguments against Mill’s account of arithmetic. Frege responds
to the claim that mathematical statements are reliable by virtue of their
acquisition from inner consciousness or sense-experience by arguing that Mill’s
account lacks implementability and cohesiveness; certain numbers are either
omitted from Mill’s notion either by their being too taxing to prove
empirically, or are altogether left out insofar as no physical facts could
conceivably correspond to them (0 and 1). Frege argues against Mill’s claim
that proving mathematical statement a
involves providing the necessary empirical basis for stating a as valid, by arguing that it is
unlikely that the truth of the laws of arithmetic are contingent on the
psychological capacity of human beings to conceive of them. Frege’s most
crucial argument against Mill, however, is concerned with Mill’s purported
confusion between the applications of mathematical statements and their
intrinsic logical sense; while Mill holds that the statement 1=1 could be wrong
pending on its validity in relation to an observed physical fact, Frege argues
that Mill is blind to the idea that there is a strict logical sense in which
1=1 obtains its meaning, and that this sense supersedes all the empirical
accidents that would make it otherwise.
Work Cited:
Frege, Gottlob. The foundations of arithmetic; a
logico-mathematical enquiry into the concept of number.. 2d rev. ed.
Evanston, Ill.: Northwestern University Press, 1980. Print.
Mill, John Stuart. A system of logic,
ratiocinative and inductive being a connected view of the principles of
evidence and the methods of scientific investigation. 8th ed. London:
Longmans, Green, Reader, and Dyer, 1900. Print.
[1] It should be noted that Mill
uses this second ‘linguistic’ account as the basis of much of his exposition
and not the ‘a priori’ account; Mill
states early in the chapter that those championing an a priori account have ‘the burden of proof’ by virtue of his
thorough examination of their positions in chapters prior. (166)
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