Download 013 Prima facie argument against formalism

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Prima facie argument against formalism
There are difficulties too with the statement: “It is an empirical fact that all of mathematics presently
known can be formalized within the ZFC system”1.
There is a problem with this claim as to an
“empirical” fact and its validity may be seriously doubted. I have never read a single mathematical
text that is written in first-order set theory (ZF or ZFC). As an empirical fact this claim appears to be
refuted by every textbook, monograph, paper and communication, written or verbal of every
mathematician, living or dead!
It may be that some of these books are about structures that are
defined in first-order set theory, but the books themselves are not written in that theory. In these
books it is the meta-text, the gloss, that is the mathematical text, which is written in a mixture of
second-order logic and natural language. Not only is the gloss that is written in this way, but the bulk
of the proofs too. Consider the following theorem – to which I append one line taken from the proof: -
3.1 Theorem
If f : a,b  
is Riemann integrable over I  a,b  , then there exists a sequence of
partitions Pk  , Pk   , such that Pk 1 is a refinement of Pk , k  1,2, ... , and
lim S Pk f   lim S Pk f   A f  .
k 
k 
Proof
... Now for any two partitions P and P  , the set P  P  yields a partition Q that is
the common refinement of P and P  ...2
One does not need to understand this extract “mathematically” to see that it is not a first-order
statement.3 The expression Riemann integrable denotes a second order property of functions; a
partition is a finite set of numbers, but we are quantifying over sequences of these, so over sets of sets,
which are equivalent to a second-order property; we have a relation between partitions (first-order
sets) of refinement – so that relation is second-order. In the proof we see that it progresses in natural
language into which are interspersed various first-order symbols; I would like to know what the
ontological status of a “common refinement” is.
There is no prima-facie empirical evidence
whatsoever to suggest that human beings think in first-order set theory. We work things out through
meanings!4
1
This is a statement made by Henson and quoted by Bringsjord and Arkoudas [2006] in Olszewski et al. [2006]
p.72.
2
This is an extract from p. 43 of Carter and van Brunt [2000] which is an introduction to the Lebesgue-Stieltjes
Integral. I choose this example by randomly picking up a book on my desk, randomly opening it at any page
and copying the first result I saw.
3
Second order statements quantify over properties, first-order statements do not.
4
I am certainly not alone in making this assertion. For example, in Mayberry [1994], we have a critique of the
“curious doctrine” that “mathematical logic is to be identified with first-order logic”. Mayberry claims that the
theories of topological spaces, Hilbert spaces, Banach spaces, Noetherian rings, cyclic groups (etc), are secondorder theories. The eliminatory theories of arithmetic, geometry and analysis are also second-order theories.
He remarks, “First-order logic is very weak, but therein, paradoxically, lies its strength. Its principal technical
tools – the Compactness, Completeness, and Löwenheim-Skolem theorems – can be established only because
first order logic is too weak to axiomatize either arithmetic or analysis.” (p.411) He outlines the properties of
The prima facie alternative to formalism is the traditionally older philosophy that symbols are
signs or tokens denoting concepts, which may also be called meanings or intensions. This philosophy
claims that when we operate formally with symbols we do so on the basis of our understanding of
those meanings – so it is the meanings that constrain the manipulations. It also claims that it is
impossible to conceive of mathematics independently of the human endeavour to understand.
Subsequently, I shall develop this into a neo-Kantian philosophy of mathematics. For the present I
wish only to introduce it.
This is part of a longer thesis advancing a refutation of strong AI. To download the thesis visit: Poincare’s thesis
For an introduction to the work as a whole visit: Introduction to Poincare’s thesis by Peter Fekete
second order logic, which is, nonetheless, “a powerful tool of definition: by means of it, and by means of it
alone, we can capture mathematical structure up to isomorphism using simple axiom systems.” (p.412)