[The foundational problem] cuts deeper than the ordinary choice between Platonism or realism, conceptualism , formalism and intuitionism … . [The] underlying question … often seems to be whether a credible integrated mathematical ontology, truth value semantics, and epistemology can best be given within a philosophical framework that interprets mathematics as ultimately mind dependent ,or as ultimately mind independent. [The] criteria by which any package of mathematical ontology, semantics and epistemology is to be judged better or best, or even as marginally acceptable … are still very much in dispute.
As a working mathematician, I tend to share many of my colleagues’ view that logic (including the topic of this work) may be important educationally, but pragmatically it is of no relevance to real-world problems. Works such as this, then, are read ‘for interest’, not for any practical benefit. On the other-hand, it may be (as some non-mathematical colleagues have politely noted) that, like scientists and other experts, mathematicians tend to be under-educated to contribute to modern life, and that we could benefit from better education, including this sort of thing. My own experience has led me to doubt that conventional pragmatism is always a good guide to life, so even though it may be the case that working mathematicians very rarely have the opportunity to benefit from this kind of work, it may still be vital that mathematicians as a community should be broadly educated and routinely debate the relevance of these ‘foundational issues’ to their own work.
At one time conventional geometry was debated. In what sense were the axioms ‘mathematical’, and in what sense subject to falsification by Physicists? Currently, I find the representation of uncertainty much debated. In what sense are the axioms ‘mathematical’, and in what sense are they falsifiable, and by whom? This anthology is not about uncertainty of probability as such, but it does give some views about what makes a for proper theories, by which we can judge views about uncertainty and probability.
Very roughly, realism is the view that there exist real abstract mathematical entities that truly have the properties they are discovered by mathematics to have. Conceptualism [holds] that mathematics is ultimately an investigation of the formal properties of ideas or concepts as the concepts of thoughts. Formalism … characterizes mathematical truths as purely formal matters of mathematical notation … .
… For the extensionalist, an object must exist in order to be referred to and have properties truly predicated of it.
… Intuitionism [is] a special … interpretation of conceptualism. [It ‘intuits’ some of the basic rules.] Intuitionistic mathematics tries formally to accommodate specifically philosophical ideas about the nature of mathematical truth and the conditions under which mathematicians and philosophers can speak even hypothetically about the truth value of mathematical propositions. [It] is not the case that every proposition of the system is either true or false, if the propositions in question have not actually been either proven or disproven. … Intuitionism is … unwilling to endorse any mathematical proposition that is not rigorously supported by concrete, surveyable mathematical accomplishment, as opposed to what it sees in the realist approach as epistemically unjustifiable projections of mathematical truth and falsehood into what is actually unknown.
[Working] mathematicians as a rule are uninterested in the kinds of questions raise by philosophers of mathematics. … Mathematicians … are by and large interested in … the theorems and relations among theorems that can be derived from interesting axioms about interesting mathematical objects. … Philosophers are exercised by questions of meaning … .
Part I: The Realm of Mathematics
Various papers seek to supplement logicism, the attempt by Frege, Whitehead and Russell to reduce mathematics to logic that Gödel showed to be impossible. Dummett advocates constructivism as a candidate foundation. This is similar to intuitionism, but requires that all mathematical objects be constructed before they can be assumed to exist. Mehlberg proposes a pluralist variant of logicism, in which every mathematical theory is reducible to some logic, but not to some universal logic.
Steiner discusses the boundary between mathematics and the natural sciences.
[Mathematics] frequently concerns itself with the problem of explaining a characterizing property of explaining a characterizing property of an entity or structure in the subject of a mathematical proof.
Part II: Ontology of Mathematics and the Nature and Knowledge of Mathematical Truth
Occam’s razor leads some philosophers to suppose that we need not regard abstract objects including mathematical objects as real existent entities, but only as psychological constructs, and as a consequence, to limit mathematical truth in what can be said to be known about mathematical entities to what has actually been demonstrated in a sound mathematical proof.
Part III: Models and Methods of Mathematical Truth
Axioms are assumptions for which a mathematical proof does not attempt any sort of formal justification, but are taken as given or posited for the sake of the conclusions they entail, or as self-evidently true. An axiom system nevertheless remains open to criticism and rejection or revision, and is ideally chosen to reflect basic definitions of concepts and truths relating mathematical objects whose principles are axiomatized. The axioms invoked in a classical mathematical proof can be considered as obviously correct on reflection, or simply postulated for consideration, or stipulated as part of the meaning of key mathematical concepts. Or, they should be understood hypothetically, and as such dispensable in principle if they should turn out to imply unacceptable conclusions, whereby mathematical proofs acquire something comparable to the status of crucial experiments in empirical sciences, as a way of testing and partially confirming or rejecting a choice of axioms.
Part IV: Intuitionism
the only method of establishing a mathematical proof is by rigorous mathematical proof.
The implications of this have been most deeply explored for number theory and set theory, and in particular in denying Cantor’s diagonalization method.
Blais proposes a three or gap-valued logic as an alternative to intuitionism, addressing many of the same concerns. He calls this ‘pragmatism’, whereby one takes an intuitionistic approach wherever possible, but allows a realist approach where necessary.
Part V: Philosophical Foundations of Set Theory
This is mainly about set theory. Much of this seems to me to be trying logically to justify the foundations of mathematics so deeply that they are necessarily circular. Of more interest is .. .
John Mayberry’s What is required of a foundation for mathematics
This is ostensibly about pointing out two things:
- ‘The set of all sets’ is not a mathematical structure, but the ‘sea’ in which structures are identified.
- One cannot both have an axiomatization of set theory and justify the axiomatic method in terms of sets.
But some of the insights are more general. (I may be biased in this, as John was my tutor at Bristol.)
John emphasizes rigorous proof and the axiomatic approach.
The axiomatic method is a method of definition. And what a system of axioms defines is a class, or as I prefer to say a species, composed of mathematical structures. Each such structure consists of a set or sets equipped with a morphology. The morphology typically consists of operations, functions or relations defined on the underlying set or sets of the structure.
The central dogma of the axiomatic method is this: isomorphic structure[s] are mathematically indistinguishable in their essential properties.
A sentence of such a logic is logically valid if it is true under all interpretations of the language, that is to say, true in every mathematical structure of the type determined by the language.
No axiomatic theory … can logically play a foundational role in mathematics.
The universe of sets is not a structure: it is the world that all mathematical structures inhabit, the sea in which they all swim.
Logic and Mathematics are generally regarded as being more certain than other forms of ‘knowledge’, but:
- We need to distinguish between mathematical claims (‘truths’) and non-mathematical claims expressed in a mathematical language. The latter may not be so deserving as being regarded as ‘certain’.
- Even within mathematics proper, there are distinctions between the most rigorous (intuitionistic) mathematics, and the less so.
For example, we can either regard geometry as a branch of mathematics, in which case it is one of the most reliable types of knowledge, or as mathematized Physics, in which case being expressed mathematically and using the most careful methods of proof does not make it ‘true’. There is also ample scope for confusion if some people regard geometry as mathematics, others as Physics. One could easily overlook the fact that its accuracy had not been checked.
The discussion in Part III does not seem to me to be quite clear. As mathematics, geometry is defined by its axioms, and cannot be falsified by Physics. As Physics (couched in mathematical language) the axioms of geometry were ‘obviously correct’ postulates, even after millennia of reflection, but were then falsified empirically. But the text of the two geometries were identical, so the nature of the axioms is surely not a mathematical property or even any kind of property of the axioms, but rather a property of the way they were regarded.
I have never before thought of applying Occam’s razor to mathematical structures. Here it is interpreted as not assuming the existence of structures that cannot be constructed. An alternative interpretation would be to assume the simplest structures, such as the ‘real numbers’, computable or not.
Here ‘pragmatism’ is considered to be being as rigorous as one can be. More often, I think, it means being as slap-dash as one can get away with. (The difference may be significant.)
Uncertainty and Probability
While probability theory is not dealt with explicitly, one can ask ‘can my probability notions be properly mathematized?’ A realist would have to contend that probabilities have ‘objective’ properties, as idealised aleatory probabilities, such as the chance of winning a lottery, do. Subjective probabilities (such as ‘the probability that my family are all well at this instant’) is more problematic. Keynes discussed the notion of ‘logical’ probabilities that go part-way towards justifying ‘objective’ subjective probabilities, but are no means complete. Irreducibly subjective probabilities could occur as straightforward numeric values in a mathematical model of someone’s thoughts, which may happen to follow ‘the rules of probability’, but these would be psychological rather than mathematical rules.
Conceptualists would see probability theory as a straightforward body of concepts about uncertainty, much like geometry is about space. But they could equally well conceptualise Keynes’ alternatives and would not necessarily see any particular theory as ‘correct’ in a broad sense. Thus one could see the broad body of work as conceptual, in this sense.
Formalists are prepared to work with any formalism of anything. But they have no way of ‘assessing’ irreducibly subjective probabilities: they can only work with what has already been formalised.
Intuitionists and constructivists deny the universal existence of truth-values, and would deny the existence of irreducibly subjective probabilities.
Whichever school of thought one belongs to, some probabilities (including adequately described aleatory probabilities) are ‘proper’, while some (including irreducible subjective probabilities that have not been formalised) are not. In practice one can generally make some explicit assumptions that enable uncertainty top be mathematized, but these assumptions may themselves be contentious ands should not be hidden.
It follows from some notions of rationality that if , for you, P(X) > P(Y) then you should prefer to bet on X happening rather than Y. This seems reasonable if P(X) and P(Y) are both aleatoric. But suppose that Y is the event that a slightly biased coin comes up ‘Heads’, for which P(Y) = 0.49, whereas X is the event that a coin, possibly double-sided, come sup heads, for which – by the principle of indifference – one has P(X) = 0.5. Then it is by no means clear why you ‘should’ prefer to bet on X rather than Y, and many would prefer to bet on Y. Thus:
Notions of rationality that include the principle of indifference need justifying.
More generally, a mathematical structure that encompasses probability must include some rule for determining P(X), and by symmetry it must be P(X)=0.5, so any such notion of rationality would have to be a theorem, not an axiom. (My own view is that this notion is not universal, and would need to be significantly bounding to be valid.