Referring to this question on the significance of the division of senses between inner and outer.

Can one say that mathematics is discovered by introspection or to coin a new word, extrospection?

Generally one thinks of introspection as appertaining to one inners mood and purposes, one is communing with oneself. Or perhaps with a god or some other indefinable inner other.

Here, I'm using introspection in a hopefully obvious way, that is almost synonymous with thinking, or perhaps seeking inspiration or illumination from the spirit of mathematics when thinking has ceased, that is in the methodical manner. I'm saying introspection to bring out that one is thinking oneself with no external other, no external world to take a sounding off. But an internal landscape, an uncommon phantasmagoria.

It may look where I'm using the word introspection I should be using the word invention. That is one side of the two-sided Janus-like dichotomy - invention/discovery. But, here I am using the word introspection as opposed to invention as the emphasis is on the interior, the psychological and the spiritual. Whereas invention has a practical, hard-headed, exteriorising and engineering kind of sound.

Extrospection should be seen as the dual of introspection in the obvious kind of way...

  • A reflection on this issue can benefit from the reading of Godel's papers about philosophy of mathematics; see Richard Tieszen, After Godel: Platonism and Rationalism in Mathematics and Logic (2011). Feb 17 '14 at 16:02
  • Perhaps you could write at least a brief answer on the basis of you reading of that text? I'm not in a position to look up that text, unfortunately - not having a university library near me. Feb 17 '14 at 16:10

Richard Tieszen, After Godel Platonism and Rationalism in Mathematics and Logic (2011), is a detailed study of Godel's late philosophical ideas.

According to Hao Wang (Godel's disciple) [page 1] :

Before 1959 Godel had studied Plato, Leibniz, and Kant with care: his sympathies were with Plato and Leibniz.

The following extract from Tieszen' book are interesting :

In the later part of his career Godel thought that finitistic formalism, intuitionism, and other forms of constructivism were inadequate as foundations of mathematics and logic. He also thought that the views of the logical positivists, whose Vienna Circle meetings he had attended, were inadequate. [page 20]

In light of [his] platonic rationalism, one could read the first incompleteness theorem for PA as follows. The first incompleteness theorem suggests that the abstract concept of objective arithmetic truth transcends our intuition (or constructive abilities) at any given stage, [...] The concept of arithmetic truth then appears to be known as an identity (or “universal”) [...]. This identity (or “universal”) is “outside of ” or “independent of ” each particular intuition (construction). This is how mathematical platonism is often characterized; that is, as claiming that there are universals or invariants that transcend the mind or our intuition. [page 47]

Godel holds that even the completeness proof for predicate calculus depends on the application of platonic rationalism. He says that the inability to find the completeness proof, although in 1922 Skolem was very close, came from a lack of the required platonic attitude toward metamathematics and non-finitary reasoning. [page 48]

The first published expression of Godel’s mathematical and logical platonism appears in the 1940s in “Russell’s mathematical logic” [page 49]

Godel [in his critique of Carnap's views] argues that not only does mathematics have content but its content is unlimited in the sense that outside of every axiomatic system that formalizes mathematical truth there exist propositions expressing new and independent mathematical facts that cannot be reduced to symbolic conventions on the basis of the axioms of this system. [...] One of the drafts of the Carnap paper contains a passage that is very important for understanding how Godel thinks of the analogy between sense perception and rational intuition. The view here is very similar to some of Husserl’s ideas about the analogy between sensory and rational intuition [page 59]

Godel’s [in the Gibbs Lecture of 1951] argument is that if mathematical objects are our own creations, then evidently integers and sets of integers would have to be two different creations, the first of which does not necessitate the second. In order to prove certain propositions about the integers, however, the concept of set of integers is required. Thus, in order to find out what properties we have given to certain objects of our imagination we must first create certain other objects. Godel finds this to be a very strange situation. In other words, we have given properties to certain objects (since they are our creations), but then, in order to find out what these properties are, we are required to create certain other objects. One might already ask how we can create properties that are, in effect, hidden from us. We supposedly create the properties, and yet they are hidden from us and we can only access them if we create certain other kinds of objects that allow those properties to be revealed. [page 67]

Godel's ideas are far from building a complete "philosophical system", but they are indeed the most recent and most authoritative view of the "realist" conception about the mathematical objects.

Tieszen's book is very detailed and go in deep into Husserl's inspiration for Godel's reflections.

  • It does make for interesting reading. I'd be interested in finding out about a little more about Husserls analogy between 'sensory and rational intuition' - it sounds similar to some of the things I've thought on, but in a vague way. Cohen mentions in an expository paper about Skolems close encounter with the incompleteness theorem, and Godels critique of Skolems reluctance to accept meta-mathematics - that is treating mathematics itself as an object for the mathematical intuition accords with Cohens view on his own 'psychological fear' of the same. Feb 17 '14 at 18:15
  • There are more books and papers of Richard Tieszen regarding the interplay between Phenomenolgy, Intuitionism and philosophy of mathematics : Mathematical Intuition: Phenomenology and Mathematical Knowledge (1989) and Phenomenology, Logic, and the Philosophy of Mathematics (2005). Feb 17 '14 at 19:49

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