ScottRoberts wrote: ↑Thu Mar 18, 2021 1:45 am
From this I can't see how, for example, one could use your system to develop a model for planetary motion. Or what the rules are for combining the symbols '<', '>', '[', and ']' into what in formalist math would be called well-formed formulas, and how they should be interpreted. Without more explanation I don't see how a mathematics with this foundation could be applied to, say, balancing a checkbook, or giving the formula for modelling the path of a cannon shot.
Thank you so much for your critical attention!
Yes, I fully agree that a sound formal foundation needs to be fully supportive of constructive play and work, the manipulation of symbols and their inter-relations in a meaningful way. The area of rationally computable discrete mathematics is of course not excluded, it can be derived from this foundation without axiomatic metaphysical postulation. I find it useful to think of the discrete area with exact numerical values as the
definite area of mathematics, and the continua as the
indefinite area. These form a complementary relation, with lots to explore of their interrelations. My math hero in the definite area is Norman Wildberger, who has been making major advances in rational and computable mathematics. A very apparent and important interrelation is computation theory, the indefinite area corresponding to temporal computation processes and the definite area to numerical data and numerical values computed. I hope and trust that mereological study of the symbols and semantics offered could offer much to study of temporal structure of computation, not just classical unidirectional computation durations but also "quantum time" bi-directional durations.
My own "talent" is very limited, only "patient" (ie. really lazy!) leisure giving room and time for intuitions to emerge and translating what appears translatable into philosophically coherent and communicable foundational language. The hard constructive work is not really my thing, I don't have the skills and patience for that. Of course I've dabbled a bit, found some string manipulation rules that seem to interact in an interesting and perhaps meaningful way, etc. Inventing and defining new rules, playing and experimenting with them and looking for perhaps meaningful generalizations that could connect with definite mathematics, reveal new areas for study, etc. is an open field for anyone to dive into.
As you ask about planetary motion, I very much agree with Norman that pi is not a number, he calls pi "metanumber" and "landscape". In my language pi is not a number either (by number we mean numerical signs that can be written), in Finnish the more general term is 'luku', which means also reading. The "Landscape of pi" is something that the new language seems to have good potential to shed new light to, my little exploration in that respect managed to reveal e.g. six basic analogues of Stern-Brocot tree written in the Relop language (a working name of the foundational theory, derived fro Relational operators). Those S-B analogues seemed to have some deep connection with polynumbers (Wildberger's analogue of polynomials), but that's where my skill limitations started to hit in. .
But aside from these concerns, I fail to see why you think this approach has any political or metaphysical significance. I understand why one might prefer to be an intuitionist rather than a formalist in terms of foundations, but nothing really follows from that, other than the former can ignore all Cantorian theory. What I am getting at is that mathematicians were happy assuming continua and discrete numbers, and doing great mathematics with them. Then some mathematicians became concerned with foundations after non-Euclidean geometry was discovered, and they were concerned with the validity of infinitesimals. And some of them worked out how one could derive continuity from the natural numbers. An interesting exercise, but it didn't really change anything. So, apparently, you have worked out how to derive the natural numbers from continua. Also an interesting exercise. So we have two ways to get back to where we started, namely using calculus to figure out planetary motion, and how to calculate probabilities, and so forth.
I do claim that the new foundation offers solution to the foundational crisis of mathematics, in which the Brouwer-Hilbert controversy is just a chapter, but which really goes back to Newton, Leibnitz and Berkeley. And that it also puts Gödel's proof in sensible context. When we consider open interval as more fundamental than closed interval, we can much more naturally agree that mathematics is inherently open and evolving system - or spirit -, not closed one, such that could be complete. In other way, the completeness hypothesis is denied also by undecidability of the Halting problem, as that is further qualified and extended by Church-Turing thesis and Curry-Howard correspondence.
Current theory of computation does not offer an explanation how computation can actually happen as continuously running currents. Theory of unconstructable real numbers can't offer that. With coherent theory of temporally self-reflecting computation, continuous durations of doing math, perhaps also a genuine AI would become a possibility. Some might like that, some might oppose that, and from the possibility it does not follow that it should be done. But knowing our curiosity, some might oppose the mere possibility so strongly from their ethical opinions that they might oppose any and all mathematical foundations that might create such possibility.
These are not easy questions.
What I don't understand is why you think the first exercise somehow contributed to colonialism. Why it is "bad" and the other "good". It seems to me that the only people who might be psychologically affected by preferring one way or the other are those who engage in one or the other exercises. And that is very few people. Certainly the sixteenth century colonialists weren't.
OK. Yes, it would seem that intellectually and theoretically, first a some sort of animistic version of idealism should be presupposed as a frame where such discussion would make any sense. In mereology of continua-durations we can think of those as layers of awareness (defined as a continuum). This layering allows further analysis of degrees of integration and separation, instead of just dissociated alter vs. M@L.
Empathy can thus be analyzed as a kind of a layer or layering of integrated awareness. In the sense of direct and continuous emotional connection, and not just imaginal identification with other, empathy means that if you hurt other, you feel some degree of the pain you cause. Discrete mathematical foundation of increasingly technocratic society comes with stronger ability to cut and discontinue layers of awareness constructing direct emotional empathy. There is plenty of empirical evidence supportive of this theory. Making good soldiers out of humans is not easy, to de-sensitize a good soldier able to kill requires very brutal intentional conditioning in the army training, and/or very brutal history - and the Crisis of the Late Medieval Age was absolutely brutal social collapse of the Feudal systems as its primary mode of production - Roman style agriculture - collapsed.
Desensitizing has been necessary feature of evolution, to be able to hunt, etc., and has involved careful and continuous search for a good balance of awareness, the dynamics between empathy and empathy barriers. My diagnosis and thesis is that combinations of spiritual and psychological layers of discrete mathematics, technocracy and brutal history have been contributing to the historical phenomena of Eurocentric colonialism, which we are now more and more seeing badly out of balance and trying to make a turn towards more empathic awareness.
Metaphysically there is the question of being (discreteness) vs. becoming (continuity), and one can complain that Western philosophy has been dominated by privileging the former. But Whitehead got past that, nevermind that he was co-author with Russell of the Principia Mathematica. I fail to see why one needs the "good" mathematical foundations to make that move. (Of course, I see the two as a polarity, but that's another step.)
Ah. In our language also being is less discrete and more continuous. More field like than object/particle like. Very clearly so in the asubjective verb in indefinite person 'Ollaan.'
Yes, limitations and conditionings of languages are not absolute and deterministic. I consider Relop just a step in a long process, building on contributions of Whitehead etc. etc. The deep ethos of mathematics is that it strives to communicate more and more clearly and confidently, creating languages of very high degree of consensus. That process can seem very slow on human scale.