Formalism is a family of views on the nature of mathematics which share a common framework.

The position in classical Greece in relation to the divisions of mathematics and the nature of proof are outlined.

Concerning attacks on the constructive ideal by Jesuit commentators of the late Middle Ages and Latin Renaissance, and the defence against these attacks.

The framework consists of five elements as follows:

1. revision of the traditional division of the mathematical sciences and their priority
2. rejection of the classical conception of mathematical proof and knowledge
3. a change in the conception of rigour
4. advocacy of a non-representational role for language in mathematics
5. the freedom to create instruments of reason

I was rather surprised by this list, which seemed at best tenuously related to what I thought formalism to be. Having read the whole article, I now understand better what he is talking about here, but I don't find him convincing on this and would not follow him in this characterisation of formalism.

The claim is that classically the division was into geometry and arithmetic of which geometry was thought more fundamental, but that the formalists inverted the priority (nothing further in this section about them actually changing the divisions).

My understanding was that even in classical Greece, the priorities changed, arithmetic having been thought more fundamental until the discovery of incommensurables after which it seemed more plausible to give priority to geometry.

Then, from the middle of the nineteenth century we are told that the formalists reversed the priorities and that this is the first element of the formalist framework.

The classical notion of proof is genetic and presentist.

"genetic" means, based on definitions and methods of construction which provide a causal explanation (note however the diversity of the classical conception of cause, and it is Aristotles formal causes which are at relevant here).

The idea is that you observe the construction and see that it meets the requirement of the definition.

This notion of proof is rejected by formalists.

The conception of rigour in classical proofs is presentist. This means that throughout the proof the geometrical construction is presented to the intuition which confirms its correctness.

By contrast the formalists emphasised abstraction from rather than immersion in intuition and meaning.

Detlefsen traces the formalist advocacy of a non-representational role for language in mathematics back to Berkeley, who by observation of the work of algebraists with ideal elements came to the conclusion that some uses of expressions can be effective and justifiable independently of any semantic content.

Some formalists have claimed the right to introduce new instruments of reasoning, This has been used to counter claims that formalists reduce mathematics to mechanism.

Aristotle (probably preceded by Pythagoras) defined mathematics as the study of quantity and distinguished between discrete and continuous quantities, which resulted in a division of mathematics into arithmetic and geometry, the sciences of multitude and magnitude. From the discovery of incommensurables, magnitudes which were not ratios of numbers, the science of magnitude was considered more fundamental (though not universally in classical Greece, Aristotle considering arithmetic more fundamental). All numbers and ratios corresponded to magnitudes, but not all magnitudes corresponded to a number or ratio of numbers (numbers for the Greeks were just the positive integers).

Detlefsen calls the ideal of proof in classical Greece the "constructive ideal", which had a genetic and a constructional component.

On the genetic component Aristotle says:

We suppose ourselves to possess unqualified scientific knowledge of a thing, as opposed to knowing it in the accidental way in which the sophist knows, when we think that we know the cause on which the fact depends, as the cause of that fact and of no other, and, further, that the fact could not be other than it is.

[You would not guess from the context of this quotation in the book that it immediately leads into material espousing the syllogism as the best method of proof. Though this is Aristotle's main message here, the syllogism is nowhere mentioned by Detlefsen as a standard of proof. This may be realistic, it is unlikely that many proofs even in mathematics, were formally syllogistic. But if we doubt that Aristotle's views in relation to the syllogism represent standard practice, why should we believe the cited passage gives a widely accepted and practiced standard of proof?]

In the case of mathematics it is the formal cause which is relevant, and the formal cause of a mathematical object is its definition.

[Syllogistic demonstration is one way of establishing this kind of causal relationship, the premises of a syllogism being regarded as the causes of its conclusion.]

The constructional component is an ideal for the realisation of the genetic component which specifies that the best kind of definition for a geometric object is one which indicates how it is constructed.

[Detlefsen's material here is not wholly convincing on the matter of this constrctional component being a widely accepted standard of proof going beyond certain kinds of geometric constructions.]

For a number of reasons trust in intuition declined. In the seventeenth century defects in the rigour of some of Euclid's geometric proofs were discovered. Assumptions not declared as axioms or postulated were found to be necessary. Some of the proofs involved tacit exploitation of elements of our intuitive grasp of geometric figures. Lambert called for proof to be based solely on algebraic characteristics in asbstraction from the subject matter, i.e. for intuition to play no role.