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3 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)