This is a short follow-up to this post https://lawrencecpaulson.github.io/2023/11/01/Foundations.html

To define the foundations of mathematics mere symbols are not enough.

We have to consider

  • the actual existence of what we call “reality” or “Universe” which has some observable properties.
  • the “Mind (of an external observe)”, which actually builds and maintains a “map” (which is not a territory).
  • the actual existence of a language-based shared human culture which stores the “previous results”
  • invention of a symbolic notations and formal methodologies for creating “system”

So, yes, intuitionists were right, and yes, the set theory is jsut another system, generalizing from what the Mind observes.

Both Sets and Natural Numbers are proper generalizations of the mind of an “external” observer.

The next obvious step was generalizations of generalizations, and this is where all the problems and paradoxes arise form.

So, does this really mean that all the non-intuitionisic views are just bullshit? Well, almost.

Ask yourself, are all the cultural artifacts – paintings, songs, literature, whole cathedrals is just bullshit?

The answer is the same for both questions. We have to parse and analyze all the socially-constructed and preserved “stuff” and trace everything back to What Is.

But what actually Is and what does not? To answer correctly this is to “remove an external observer” and to separate clearly the “notions of an observer” from “where they came from”.

It is to recognize the fundamental difference between a capture by the mind and a proper generalization into an abstract concept (and then studying its inferred properties), from what has been “observed”.

So, Natural Numbers, The Right Angle (especially the notion of a perpendicular) and most remarkably - Sets, are the proper generalizations.

Sets are special because they are what an observer “sees” when one looks “at the front of reality or Universe”. A simplified metaphor would to see an actual tree as sort-of “sphere of leaves”, where in reality each leaf is attached to the thunk.

The underlying tree-like structure of Causality, which can bee seen (by an observer) as an acyclic directed graph, is what “causes” the “classes” and “sets” and any structure whatsoever (both actual and abstracted out).

So, what are the Foundations of Mathematics?

The foundations of mathematics are these properly captured generalizations. They are literally the “starting points” or, well, a foundations.

The full list of these proper generalizations (especially the task of separation them from mere abstract bullshit) is a major task, so there are just some “illustrations”.

The notion of “putting together” is proper, which is the basis of addition and commutativity (it is irrelevant which part was the “first”. There is no first).

The notion of an infinity is not, because it is based on an abstract mental process. Yes, another +1 could always be added, the whole construct there is nowhere to be “seen” outside your head.

Infinities are mental and then social constructs. The notion that one infinity is denser than the other is, again, merely an inference.

The ability of the mind to superimpose a “ruler or a scale” and count the notches is a proper generalization, even to the point that in a 3D space any “cup” can be a measure of a volume.

The Cartesian coordinate system, for which an Origin could be placed anywhere and a Unit can be of an arbitrary length is no different in its nature from Sets and Natural Numbers.

The notion of a direction, distance and angle give us vectors, which, of course, are also proper generalizations.

Notice that all the formulas related to these proper generalizations are discovered, just as the law of physics (which describe the actual “forces”).

What about all the Platonist’s? Well, it is a church and a sect (a social movement based on abstract bullshit, literally). All the Platonist’s artifices are no different from religious artifacts stored and preserved in the shared culture.

So, was it all bullshit? Again, just as in the case of organized religions we have to parse and analyze.

Upanishads and the Buddha have produced lots of proper generalizations in the realm of the “mind itself” and of “the social”.

In short, The Properly Generalized Abstraction and their derived properties are the foundations, and everything else is socially-constructed bullshit.

There is no Mathematics without the Mind, the Environment (Universe), and a language-based shared culture.

This, by the way, corresponds to the popular meme-view that “we live is a simulation”. The principle is that we will never be able to tell due to principial impossibility to penetrate through the actual “abstraction barriers”.

Real mathematics is, thus, a study of observable abstraction barriers and their derived properties.