Of the Nature of Mathematics

Define invention. Sorry, I just need to be clear on what is meant by invention. If it means making up something that isn't real, then I disagree.

No apologies; you are correct. Some qualification is in order.

By 'invention' I mean "...making up something" that "is" real, rather than non-real. Mathematics, as the pure quantitative science in tandem with formal logic, would be, in this sense, a make-up of something real, since it expresses the world as such correctly (quantitatively, that is).

Does this help Ray?

Yeah sort of.

I think it's a bit of both. The relationships are there, and we express it in terms of mathematics.

The status of mathematics has always intrigued me.

In a sense it is an "artform" and therefore could be classified as "invention", but its applicability to "the world" as a coherent model can lead to "discovery", or restructuring of observations.

In general we must be careful with the word "reality". The lowest level of measurement is "nominal" or naming. This is already a projection of "structure" on the world by selective classification. Similary we must be careful with "invention" since Prigogine has shown that "structure" can arise "spontaneously" in dynamic systems, whether these are considered internal or external.

Herman Hesse's fantasy novel "The Glass Bead Game" explores and extrapolates some of the mutualities between the evolution of mathematical and worldly structures.

Interesting. I would also include both terms but for different reasoning.

Mathmatical concepts seem to be discoveries in that they are findings of rules that already appply in the universe. Since these already existed they can't be said to have been invented.

The symbols and methodology we use to explain those findings would be inventions though IMO.

I follow Russell and Ramsey, including Wittgenstein, on the nature of maths; that is, on the relation of maths to logic, and the former as reduced to the latter.

And Fresco has provided us with a well-formed reply, to my mind: as an art-form, though equally applicable as 'invention' and 'discovery'.

So, to continue:

What is your view of the relation of maths to (formal symbolic deductive)logic?

NobleCon

Two points.

1. Piagets views on the development of intelligence have "logical processes" as an end product and in no sense an a priori with independent existence.

2. Binary logic, including the law of the exluded middle is not sufficient to describe a dynamic world of shifting class memberships.
Since "progress" has been in this direction with non-binary logic it seems that "logics" are subsystems of general mathematics.

May I ask: what do you refer to with the term 'progress'? As for logics in general, as well as their relation to "general" maths, I ask for your opinion on the following:

The rules of inference are utilized in mathematical derivations, and mathematical logic utilizes such rules, or principles. In point, maths both utilizes these rules and applies them to mathematical logic, and, in tandem, that form of logic is developed through such derivations. Maths utilizes its own form of logic, derives its own structure through this form, and derives that form through the derivation of that structure via the rules of inference. The one generates the other, and, at once, generates itself.

In what manner, therefore, can we say, with some degree of security, that non-binary logic may be represented as a subsystem of general maths? Some would hold that it is maths that is a subsystem of logic, and not vice versa.

What is your view on this Fresco?

NobleCon

By "progress" I mean successful applicability
(for example of fuzzy logic to neural nets)

I am not specifically familiar with the Philosophy of Mathematics, but I note from Google that opinions differ on its ontological status. I agree with you that there has been a dynamic interplay and evolution between what we might call "logic" and what we might call "general mathematics", but the more interesting question is whether the dynamics themselves can be characterized by second order mathematical models such as catastrophe theory.

Do you mean such second-order models to represent this dynamic interplay between the two, logic and mathematics?

As for catastrophe theory, I am not familiar with it. If you have a moment to spare, tell me about it- what is it, what does it consist of (the logical structure), and what is its objective.