of the notion formal. Formal as decoupling from semantic content and formal as computable. If you remember, I actually made a very brief mention of this de-semantification and basically decoupling from semantic content when I talked about axiomatic regimes. And I think Jessica asked me a question, and that was it, I think. So in a sense of semantic decoupling or abstraction from intentional content, so basically when we are talking about semantic decoupling, formalized decoupling from semantic content,
we mean it in the sense of abstraction from intentional content. In this sense, the notion of formal corresponds to the familiar idea that the abstraction in question concerns abstraction from all meaning whatsoever, i.e., that it amounts to what we could refer to as a process of decoupling from semantic content. to use German philosophers, Sybil Kramer's term, the semantification of some portions of written language. There are some essays I've ever translated,
but not her major book. This is the name. In this sense, to be purely formal amounts to viewing symbols as inscriptions with no meaning at all, as pure mathematical objects, and thus no longer as signs, at least not in the sense of a referential sign. So basically, symbols at this, basically, in this sense, in this kind of, you know, in the sense of this notion of formal, are pure inscriptions, or what we can call pure
and empty sign. But not sign in the sense of the kind that we talked about, namely indexical and iconic signs, because iconic sign has basically a discriminatory interpretation capacity, and indexical sign had basically a causal connection to the accumulation of a statistical inference that we talked about last session. So it's important to understand formal and basically as de-semanified being basically
the repertoire of meaningless, totally abstracted inscriptions. And this is something I will get back to this in much more detail about basically what is exactly at the logical level, what is exactly in a purely meaningless, non-referential sign that allows us to develop massive amounts of, basically, cognitive freedom. I will talk about this quite in detail when we talk about Girard's theory of low-costs,
especially the kind of logical system that allows us to develop that can have both the properties, interactive properties of natural language and formal properties of basically computational systems. So historically, variation of the notion of the formal as basically representing de-semantification, or recoupling from semantic content, has its roots in 19th century and was later consolidated with Hilbert's program on the foundation of mathematics. So as I said, this session is going
to be just like the first session of the complexity one, just a story about this emergence of computability theory. And the first point we want to basically develop is this connections from basically the notion of formal and how it basically transforms once it's being introduced to the notion of axiomatic systems in mathematics, and later on basically transforms into basically what we know basically as computability
theory. Frigge criticizes this approach, in particular as defended by his colleague, Karl Thome. And what he criticizes is this notion of formality in this sense. Responding to Frigge's criticism, Thome writes, for the formal conception, arithmetic is a game with signs which one may call empty. By this one wants to say that in the game of calculation they have no other content than that which has been attributed to them concerning their behavior with respect to certain rules of combination, i.e. rules of the game.
The gist of the formalist position defended by Thome seems to be a plea for the ontological freedom. I.e. mathematical symbols need not to be thought of as picking out mathematical objects and and properties, and thus mathematical propositions do not state mathematical facts. Later with Hilbert's program in 1920s, a variation of the sense of the term formal became pervasive, especially once Hilbert's approach was described as formalist. However, it must be a stress that Hilbert's views are significantly more sophisticated than the idea of treating mathematical symbols as meaningless.
So Heidel's program should not in any way be viewed merely as a continuation of Thome's formal conception of arithmetic. The gist of the Halbertian notion of the formal is aptly captured in the following passage by Paul Bernice. So this is the passage. The typically mathematical character of the theory of provability reveals itself especially clearly through the role of logical symbolism. The symbolism is here the means for carrying out the formal abstraction. The transition from the point of view of logical content to the formal one takes place when one ignores
the original meaning of the logical symbols and makes the symbols themselves representatives of formal objects and connections. For example, if the hypothetical relation, if A then B, is represented symbolically by A arrow B, then the transition to the formal standpoint consists in abstracting from all meaning of the symbol arrow and taking the connection by means of quote, sign, quote, arrow. It's basically the idea of this quotation comes in the formalistic school and later,
you know, so many philosophers adopt this basically device. This device, also known as quotation, basically bracketing or basically dot marking or sort of, it's just you have a dot, two dots, and then you can fill in the blank with any kind of symbol. It simply has an illustrative role. It basically connects your sign not with any basically referential fact, but it simply shows that your sign stands in a symbolic relation in the sense that we talked about
last session about symbols stand only in combinatorial connection with one another. So it shows that your sign, and in this case the arrow sign, basically wrapped into quotation mark. It shows that your sign basically has a functional rule, a specific functional rule in your symbolic system with regard to other signs in that basically repertoire of symbols, finite discrete
symbols. So as I said, then the transition to the formal standpoint consists in abstracting from all meaning of the symbol arrow and taking the connection by means of the sign, quotation, quotation arrow itself as the object to be considered. To be sure, one has here a specialization in terms of figures instead of the original specialization in terms of content. This, however, is harmless in so far as it is easily recognized as an accidental feature. Mathematical thought uses the symbolic figure to carry out basically the formal abstraction. A key term here is formal abstraction.
Abstraction is, of course, a crucial concept in mathematics. And from an Aristotelian perspective, abstraction typically corresponds to abstraction from sensible matter. In fact, this is part of the classical canon of Aristotelian basically theory of abstraction. Aristotel, in fact, developed theory of abstraction from Plato's cousin who came up with the word apharisis, meaning subtracting away. He basically developed a theory of abstraction to ontologically prioritize mathematics, namely the domain of form over physics,
basically the domain of the sensible motion. And this is basically the whole idea is that we can, at least in the Aristotelian metaphysics and theory of abstraction, we can basically abstract mathematical forms from physical sensible properties and basically from motion. But of course in Aristotelian work and theory of abstraction, this abstraction of mathematical forms from physical properties.