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Knowing the Numbers
1. Introduction
Physicist Richard Feynman was once asked why he thought he was better at
mathematics than most people. He thought about it and said, ‘Other people know
about numbers, I know the numbers’. Feynman was claiming knowledge by
acquaintance of the numbers. In fact we do not think that Feynman explained his
expert knowledge correctly. Feynman was undoubtedly familiar with all kinds of
facts about mathematics. However, it is a mistake characteristic of naive, Platonist
realism to model knowledge of the numbers on perceptual knowledge of objects. We
are not opposed to realism. However, we argue that an adequate realist theory of
mathematics must conform with the following constraint:
(NKA) We cannot know all of the numbers by acquaintance.
Some numbers could perhaps be known by acquaintance. It depends on one’s view of
numbers. But it seems undeniable that there are some numbers—typically very large
numbers—that we cannot know by acquaintance. More precisely we will be happy to
demonstrate the existential claim:
(NKA’) There are some numbers that we cannot know by acquaintance.
Quine’s empiricist philosophy of mathematics has great strength in maintaining
realism alongside with (NKA). Maddy’s realism (Maddy (1990)) denies NKA for
small numbers and sets, and essentially adopts a Quinean approach to the higher
reaches of mathematics such as transfinite set theory. Naive Platonists deny (NKA),
where ‘acquaintance’ is understood as a form of ‘intellectual intuition’. Who is right?
What follows below are three arguments against NKA.
2. Arguments against knowledge by acquaintance
Let’s illustrate our claim that (NKA) is true with reference to the real numbers.
2.1 Indefinables
One way of demonstrating the impossibility of such numbers is to consider all the
indefinable real numbers. There are vast swathes of the real number system that are
comprised by numbers lacking any discernible pattern that cannot be calculated
according to a rule. We must believe in the existence of such random reals on pain of
the real number system not behaving as we think it should. In particular, the
uncountability of the real numbers depends on there being indefinable real numbers.
This follows because the number of definable real numbers is merely countably
infinite. 1 Equally it’s obvious that indefinable reals are not (and cannot be) known by
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The argument for the countability of definable reals: you could theoretically draw up a list of such
definables by giving them an ordering (eg. a lexicographical ordering).
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acquaintance. As the indefinable reals lack any pattern, surveying a finite portion of
the decimal expansion of such a real is insufficient to extrapolate to the whole decimal
expansion. Such numbers are therefore unsurveyable.2 Moreover, by definition, such
numbers do not have names or formulas by which we can refer to them. Indeed,
singular reference to indefinables, in the sense of direct reference, is otiose. Such
features of indefinables make it extremely unlikely that we have acquaintance with
such numbers.
2.2
Part of the structure
One of the key insights of structuralism is that mathematicians are largely not
concerned with the intrinsic properties of numbers; they are concerned rather with
more general structures in which the numbers may figure as positions rather than as
complete objects in their own right. 3 Anything that occupies the appropriate role or
position in a certain structure may assume the identity of a particular number.
Acquaintance is a mode of singular knowledge of an object. Acquaintance with an
object is what makes it appropriate to refer to it demonstratively rather than
descriptively. In the term ‘this F’, the function of ‘this’ is to an item of experience (an
F item of experience given the complex term, ‘this F’). In the case of descriptions
such as ‘the F’, reference is not necessarily singular: should anything else be F, the
description will fail to refer to one particular item.
If the structuralist insight is correct, then mathematicians do not engage in singular
reference to the numbers. If the mathematician only cares about the structure of the
natural numbers up to the point of isomorphism, then none of the intrinsic properties
of the numbers (if they had any) matter. Acquaintance with an object, however,
requires presentation of the object to the perceiver. It seems reasonable to assume that
acquaintance would provide some awareness of an object’s intrinsic properties.
Therefore we can conclude that the mathematician does not aim at acquaintance with
the numbers.
2.3
Abstractness and Causality
Whatever the nature of the ‘acquaintance’ relation, it seems clear that, like perception,
acquaintance requires a causal interaction with an object. If x never interacts causally
with object o, then we can infer (if x is the right sort of thing, a conscious being that
can be aware of objects) that x is not acquainted with o. So causal interaction is a
necessary condition on acquaintance with an object.
On one traditional conception of abstract objects, an abstract object stands outside
spacetime and is causally inert. Many philosophers have assumed, following Plato,
that mathematical objects will be abstract in this traditional sense. Hence,
mathematical objects will be causally inert. Consequently, it will be impossible to be
acquainted with such objects.
One might take this line of reasoning as a reductio ad absurdum of the whole
traditional Platonic line of reasoning on which mathematical objects are abstract in the
Definitions of ‘knowledge by acquaintance’: cf Russell.
Definition of ‘intrinsic property’: F is an intrinsic property of x if x would have F without considering
x’s relation to any other object. Intrinsic properties are non-relational properties.
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traditional sense.4 It is plain that we do have knowledge (if not knowledge by
acquaintance) of mathematical objects. However, if we are willing to allow
knowledge by means other than acquaintance (knowledge by inference to the best
explanation, for example), then the complete rejection of Platonism could be avoided.
The Platonist can simply agree that we are not acquainted with mathematical objects
while disagreeing that acquaintance is necessary for knowledge.
Conclusion
(NKA) holds
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The line of argument is familiar from Benacerraf (1973), ‘What is mathematical truth?’