Edwina, Helmut, List,
Since the issue about Peirce's three universes was mentioned in your notes, I'm including an excerpt that I had intended to include in the article I just finished. (See below)
Although it's relevant to the content of that article, it raises too many questions that would require more explanation. After the excerpt below, I include two links to other articles in which I discussed some related topics.
John
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Text omitted from the article on phaneroscopy:
Plato and Aristotle disagreed about the role of mathematics. Plato claimed that mathematical forms (such as Peirce’s diagrams) are prior to any physical embodiment, but Aristotle claimed that mathematical entities are not separable from sensible things. Peirce’s three universes of discourse resolve this conflict: the possible, the actual, and the necessitated.
The universe of possibilities is the domain of pure mathematics. Every mathematical theory begins with some hypothesis expressed in a diagram or its algebraic linearization. The special sciences study the universe of actuality. The hypotheses (diagrams) of mathematics are applied to aspects of actuality in order to make predictions. The hypotheses that make reliable predictions are the laws of science. They are the best known approximations to the laws of nature. The totality of laws of nature is the universe of the necessitated.
Although Aristotle did not discuss signs in his metaphysics, his earlier writings (the Organon) covered logic and semiotic in his analysis of sêmeion, symbolon, and logos. For Peirce, mathematical phaneroscopy leads to the three categories (trichotomy) of Firstness, Secondness, and Thirdness, which classify all the signs of perception, language, and the sciences. The dotted lines of Figure 1 show the flow of diagrams and theorems from mathematics to the other sciences:
- Possibility. Every mathematical theory develops the implications of some possible pattern (diagram). There is no reason to exclude any possibility or to deprecate it as a fantasy. Some fantasies may be adopted as plans for engineering projects. They then become aspects of actuality.
- Actuality. The special sciences observe patterns in the actual universe, find and apply mathematical theories about those patterns, use those theories to make predictions about what may happen, make new observations to test those predictions, revise the theories, and repeat.
- Necessity. The propositions entailed by any pattern by any diagrammatic reasoning are necessarily true of any occurrence of that pattern. All theories of science are fallible, but the best are reliable on those domains for which they have been thoroughly tested.
All mathematical theories must be available for applications to the special sciences. All semiotic patterns are necessary for representing natural and artificial languages. In fact, every artificial language in mathematics and computer science is a disciplined application of the syntactic and semantic mechanisms of natural languages. Value judgments are necessary for reasoning about the beliefs, desires, and intentions in any social activity or organization — and the organizations must include colonies of any species from bacteria to humans or even aliens from other galaxies.
If the diagramming conventions are precisely defined, these rules are sound: observation and imagination would add duplicate information in some area; and erasure would delete duplicates. For scenes in nature, photographs, and informal drawings, these rules may be useful, but fallible approximations. For more discussion and examples, see “Peirce, Polya, and Euclid: Integrating logic, heuristics, and geometry” (Sowa 2015) and “Reasoning with diagrams and images” (Sowa 2018).
I presented the talk on "Peirce, Polya, and Euclid" at an APA session on Peirce. I later presented an extension to the slides at a workshop hosted by Zalamea in Columbia in December 2015. See htttps://jfsowa.com/talks/ppe.pdf
The article on "Reasoning with diagrams and images" is an extended version of the material in ppe.pdf. See the link in slide 2 of ppe.pdf. It's helpful to read the slides before going to the longer article.