** Rationalism** prioritises reason, holding that the truths of mathematics are ‘discovered’ by proof, and considered to be a priori truths. **Empiricism** focuses on experiment. Once a central concern of philosophy, mathematics has become so specialised as to be incomprehensible to many philosophers, although logic is central to both and areas such as set theory are of obvious interest to both. Mathematics is an obviously interesting case study, with many questions around its ontology, epistemology, semantics and logic, and its place at the heart of scientific models.

**Intuitionism** holds that theorems could be deduced by a process of introspection, and mathematical objects are mental constructions. Interestingly, they deny the validity of the law of the excluded middle. Mathematics is considered to be a study of functions of the human mind.

**Object **

What are all these numbers and properties and things?

**realism in ontology (Platonism) – **mathematical objects exist independent of the mathematician

**idealism** – mathematical objects depend on the human mind

**nominalism** – mathematical objects are linguistic constructions

**Truth**

**realism in truth-value – **statements have **objective** **bivalent** truth-values independent of subject

**anti-realism in truth-value – **truth-values are dependent on the mathematician or the structure of the human mind, and can be indeterminate

(See: Mathematical Truth by Paul Benacerraf, The Journal of Philosophy, Vol. 70, No. 19, Seventieth Annual Meeting of the American Philosophical Association Eastern Division. (Nov. 8, 1973), pp. 661-679. http://thatmarcusfamily.org/philosophy/Course_Websites/Math_S08/Readings/Benacerraf.pdf)

The Ancient Greeks messed about with geometry and constructed numbers.

5th century BC. Plato believed in eternal, unchanging, **a priori** mathematical objects that exist in what’s now known as a **platonic realm**, outside of physical reality, and so considered the use of dynamic language in mathematics inappropriate. Aristotle held that mathematical objects exist IN perceptible objects, being a bit of an empiricist, and the separation is a psychological process of *abstraction*. Frege’s pretty sarcy about that one.

17th Century. Descartes, Leibniz, Spinoza are **rationalists**, Locke, Berkeley, Hume and Reid are **empiricists. **Kant takes arithmetic to be **synthetic** (as opposed to **analytic**) and **a priori, **founded on **intuition**. For him, a proposition is analytic if the predicate concept is contained within the subject concept, and synthetic otherwise, and he does not consider that 5 + 7 = 12 could be established through examination of the concepts of 5, 7, 12, +, =, etc., because we need intuition to represent numbers (which has to do with perception or the structure of the human mind).

John Stuart Mill is an **empiricist **and a **naturalist**, holding that the human mind is a part of nature and so no knowledge can be a priori. Verbal propositions are true by definition and have no content. Real propositions (which includes eg. sums) are generalisations from constant experience from early on (Frege’s pretty sarcy about this, too). Generalisations are just summaries of empirical evidence and predictions. He takes lines, points etc to be *limit concepts. *

Hume’s Principle, written by Frege, but hey…: For any concepts F, G the number of F is identical to the number of G iff F and G are equinumerous.

Almost the 20th century, and **Logicism**, then… Frege has a different definition of analyticity (as opposed to syntheticity) to Kant, applying it to arguments that a based on logical truths as opposed to general science, and messes about with ideas about equinumerosity which seem quite useful. Russell likes Frege’s account of the natural numbers, and thinks a lot about *classes* (as logical fictions), but doesn’t like *impredicative* (circular) definitions. Russell and Whitehead try to sort everything out with new axioms and make a big, horrible book that people still think failed. In a **linguistic** turn, Carnap and **logical positivism **like the idea that the truths of mathematics are analytic and a priori, a priori knowledge now understood to mean knowledge of language use, necessary truth being truth by definition; they say mathematical truths are necessary truths but have no factual content.

(In the late 20th century, **neo-logicists** like Crispin Wright, Bob Hale and Neil Tennant maintain that 1) mathematical truths are knowable a priori by derivation from analytic rules and 2) mathematics concerns an ideal realm of mind-independent objects.)

**Term formalism **claims that the essence of mathematics is in the manipulation of characters, which aligns with some of the experienced development of mathematics eg. the introduction of imaginary numbers. Frege is very rude about this indeed and gets a bit tedious. Or, **game formalism** likens mathematics to a game. Either way it tends to sidestep a bunch of epistemological questions by pointing at the rulebook, and saying, that’s what it is. **Hilbert** got very, very into axioms, and was mostly interested in the *relations* between things, substituting beer mats for points and all sorts. He then wandered off into meta-mathematics and wanting everything sewn up nicely and proved to be consistent to address the crisis in the foundations of mathematics: *the Hilbert programme. *We must know – we will know! He’s into *finitary *arithmetic.

The in the 1930s Goedel screwed him right over, wonderfully proving that any formal axiomatic system must be incomplete or inconsistent, which is really fun, using a bunch of paradox type things, which is also really fun. Haskel Curry is interested in how branches of maths become codified, arguing that all other forms of maths are vague and full of metaphysical assumptions, and is quite **pragmatic, **being less interested in notions of truth than in “considerations which lead us to be more interested in one formal system rather than another”.

Then there’s **intuitionism**, and Shapiro’s saying that mathematics is primarily a mental activity, which I’m not sure is that certain. Intuitionism is a general term for philosophies that don’t buy into the* law of the excluded middle* (“See Dummett 1977”), so proof by contradiction goes out the window. Intuitionists are at least highly suspicious of **realism**. Brouwer is quite Kantian, both holding that experimentally disproving mathematical laws is unthinkable. Brouwer wants to found mathematics on time rather than space, and idealised mental **construction,** and sees language and logic as a “non-mathematical auxiliary, to assist the mathematical memory or to enable different individuals to build up the same [construction]”. Heyting then formalised intuitionistic logic which seems a bit mad – both see language as an imperfect medium for communicating mental mathematical construction, but Michael Dummet is more into linguistics, saying that *meaning is use *(ie. not contained in the tool, it’s the use), and that “classical mathematics employs forms of reasoning which are not valid on any legitimate way of construing mathematical statements…”

In contemporary thought, there are **realists in ontology,** who believe that numbers exist, and sceptical **nominalists **who accept the importance of mathematics but avoid invoking mathematical entities. W. V. O. Quine attacks the separation between analytic and synthetic, and **reductionism**, talking instead about systems of beliefs as a web, and he and Hilary Putnam’s *indispensibility argument *argues that the use of mathematics in scientific enterprise guarantees existence to the real numbers. Penelope Maddy defends a double realism that takes bits of Goedel’s platonism and Quine’s empiricism and uses set theory as a foundation for mathematics.

Hartry Field goes for **fictionalism**, and looking at the *indispensibility argument* but saying that numbers aren’t NECESSARY to science Field provides **nominalist**ic formulations of scientific theories (ie. making no reference to abstract objects like numbers or sets) – and separates out the *concrete* from the *abstract*. **Modal construction **is talked about by Chihara and uses open sentences in place of sets.

*Metaphysical Myths, Mathematical Practice *(1994) by Jody Azzouni and *Platonism and Anti-Platonism *(1998) by Mark Balaguer both seem pretty great but I’m not going to try to understand/summarise them here.

Finally, there’s **structuralism**, realist in truth-value but with varying ideas about the existence of mathematical objects, which says that MATHEMATICS IS THE SCIENCE OF STRUCTURE. The structuralist rejects ontological independence among numbers, arguing that they are all co-dependent, ditto systems, and the pattern or structure is abstracted from their relations. The **ante rem **structuralist says that this pattern has existence independent of any systems that exemplify it, and structures exemplify themselves, places, construed as objects, exemplifying the structure. Structures or indeed athematical objects then are not considered particularly to be OBJECTS. Different versions of structuralism have different ontologies and different epistemologies. There’s a nice bit about structuralism and pattern recognition here.

from *Thinking about Mathematics: The Philosophy of Mathematics* by Stewart Shapiro, OUP Oxford, 13 Jul 2000

I used that, smaller text, but Shapiro also edited the giant beast that is the Oxford Handbook of the Philosophy of Mathematics and Logic, an excellent and intimidating reference text that suggests that he’s a pretty good person to go to.