More/Less Mathematics

[ScottRoberts]

Dynamic tetralemma:

1) increases <
2) decreases >
3) both increases and decreases <>
4) neither increases nor decreases ><

Another shorthand for variety of such interdependent verbs is more-less.

3rd lemma connects with notions of interval, duration, and self. With open interval that continues beyond horizon, to be more exact. In Finnish interval is 'väli', and the verb 'to care' is 'välittää'. In that semantic connotation, the new foundation is not cold and abstract, it's caring. <3

The foundation starts from irreducible notion of continuum/continua, which meditation science and notion of awareness confirm and necessitate. In that sense it's common, natural and empirical.
Discrete/discontinuous quantification can be derived from the 4th lemma, which denies/halts the continuity of processes and creates ability to think also abstract states. As a necessary relation for any theory of mathematics, the notion of equivalence follows naturally: If A is neither more nor less than B, then A=B. Also closed intervals (aka finite sets) can be derived from 4th lemma: []. Combining open and closed intervals we get following basic forms which include also various beginnings and ends of continuous processes: <>, [], [<, [>, <], >]. These can be further combined, studied and their relations analyzed in mereology of durations/intervals. Mereology means study of part-whole relation. Most basic mereological relation is the Russian Doll, interval inside interval, e.g. <<>>

Of course also standard school math can be derived from this foundation, as we have already basic tools of equivalence relation and various intervals, and deriving already familiar math is fairly trivial and IMHO not very interesting. The new approach to mereology is for me much more interesting, as it enables much more mereologically rich study and discussion of e.g. M@L - alter relations.

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.

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. 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.

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.)

[SanteriSatama]

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.

[ScottRoberts]

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.

Well, I too am lazy, so I just read (well, mostly skimmed) the first chapter of Wilderberger's book, and am unlikely to pursue it further. What I am not impressed by is the motivation of presenting an easier trigonometry for high school students. As I see it, they should be challenged, since if they are going to go on to a profession that requires serious applied math, there are much bigger challenges ahead of them. That is, they are not likely to just need trigonometry.

I also don't see anything gained by saying that pi is not a number. How else to mark the halfway point around a circle? I suppose I should ask first if all the irrationals in his scheme are not numbers? What about e, or i? In measurement, "pi meters" is just as precise as "1 meter". And if not talking about measurement, pi and 1 are equally abstract.

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.

I think of Gödel's and Turing's proofs as simply showing that Hilbert's dream is stillborn, and has no further metaphysical import. Meanwhile, the openness of mathematics simply follows from the openness of imagination. In my view, there is no foundational crisis. Foundational mathematics is just another branch of mathematics. If there is an inconsistency in one's mathematics, it will eventually show up, and then one can revise.

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.

My view is that there cannot be a self-reflecting computation, or any self-reflecting mathematics. Mathematics is just form, and for reflection one needs formlessness, which cannot be realized mathematically.

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.

Here we are going to get into the Cleric/Santeri-Lou discussion, which is unlikely to be any more resolved than that. I will simply state that I side with Cleric, that the animistic mindset was a stage we grew out of, as we will grow out of our current stage. To relate to this topic, I would say that the growing out of this stage will be a matter of self-discipline, of learning to will our thinking, and that mathematics -- any mathematics, set-theory based or not -- is good training (as Plotinus held).

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.

I would just say that turning towards more empathic awareness will happen -- is happening -- because we are simply growing up, and seeking out more benign mathematical foundations will have no effect on the process.

[SanteriSatama]

Well, I too am lazy, so I just read (well, mostly skimmed) the first chapter of Wilderberger's book, and am unlikely to pursue it further. What I am not impressed by is the motivation of presenting an easier trigonometry for high school students. As I see it, they should be challenged, since if they are going to go on to a profession that requires serious applied math, there are much bigger challenges ahead of them. That is, they are not likely to just need trigonometry.

Norman is all about making pure math more pure. That does not mean he rejects applied math. I don't have the book, I've been following the WIP video series, where the really good stuff is and keeps on coming.

I also don't see anything gained by saying that pi is not a number. How else to mark the halfway point around a circle? I suppose I should ask first if all the irrationals in his scheme are not numbers? What about e, or i? In measurement, "pi meters" is just as precise as "1 meter". And if not talking about measurement, pi and 1 are equally abstract.

My point was to discuss the distinction between Finnish 'numero' and 'luku', which relates very closely with what can be constructed, ie. written down with numerals, and what can't be. In the realm pure math, Norman's core principle is what he calls 'Law of logical honesty': "Don't pretend to be able to do what you can't do'.

Brouwer-Hilbert controversy boils down to constructibility, and (in my interpretation) Norman takes a very strict principle of constructibility as the founding principle of his approach. Transcendentals etc. irrationals can be "traced" in some magical sense (three dots etc. clues for intuitive imagination), and that magic is the area to which Finnish mathematical term 'luku' refers to, as well as the concept of domains in the meaning of domain of whole numbers, domain of rational numbers. Basically anything where the magical 'three dots' shifts from strict numerical constructibility to intuitive imagination.

In my view Norman's approach is very respectable and humble example of the principle 'Let's not try to run before we learn to walk'. And also, let's see all the places we can walk to, before trying to run. His approach has been very productive walk revealing new deeply meaningful algebraic connections, which extend to hyperbolic and parabolic geometry, etc., showing the power of Whiteheadian planar paradigm constructed as quadrances and spreads. I won't go to Norman's constructions of circle, check those yourself if you are interested.

In Relop language, the following form can be interpreted and investigated as a landscape of pi:

<>< <> ><>
<<> <> <>>
><> <> <><
<>> <> <<>

Each palindromic line is a different form of basic number-antinumber relation (e.g. positive and negative numbers of Integers). "Equator" can be imagined between 2nd and 3rd line. If we count the "heads" <> with or without "tails" <>< and "beaks" <<>, each line has three heads. and some other processes going on.

I think of Gödel's and Turing's proofs as simply showing that Hilbert's dream is stillborn, and has no further metaphysical import. Meanwhile, the openness of mathematics simply follows from the openness of imagination. In my view, there is no foundational crisis. Foundational mathematics is just another branch of mathematics. If there is an inconsistency in one's mathematics, it will eventually show up, and then one can revise.

The foundational crisis has been about constructibility, the dream of a foundation for mathematical language that would be able to consistently or at least coherently describe continuous movement. Applied mathematics of calculus works to some extent in practice, the crisis since Berkeley is the missing link between applied and pure math.

My view is that there cannot be a self-reflecting computation, or any self-reflecting mathematics. Mathematics is just form, and for reflection one needs formlessness, which cannot be realized mathematically.

We've been through this before. As Gödel, Escher, Bach shows, Gödel-problematics at deepest level reflect the problem of self-reflection.

Reflection as such is a form, and self-reflection cannot be realized mathematically in the confines of any static model. Same does not apply to dynamic model, and if process of self-reflection can be coherently characterized by relations <, > and =, Relop should be able to do that. Note the dynamic "near-polarity" of almost palindrome polar-relop. Also the different letters a and the e are almost symmetries on the vertical axis. A difference of difference.

Here we are going to get into the Cleric/Santeri-Lou discussion, which is unlikely to be any more resolved than that. I will simply state that I side with Cleric, that the animistic mindset was a stage we grew out of, as we will grow out of our current stage. To relate to this topic, I would say that the growing out of this stage will be a matter of self-discipline, of learning to will our thinking, and that mathematics -- any mathematics, set-theory based or not -- is good training (as Plotinus held).

Other words for training are askesis, yoga and kung fu. In the social aspect of mathematical askesis, a certain kind of flexibility and rigour of mind is required. Conjectures are tested, and according to testing, rejected, accepted or further qualified. Important part of the practice is to study and comprehend, why axiomatic set theory should be rejected for foundational, constructible purposes. As well and what heuristic purposes it has served.

Mathematical practice is also very goof for thinking temporal relations and assumptions behind various theories. In light of palindromic time of quantum theory etc. etc., temporal character of theories of consciousness in the form State A -> State B -> State C is very limited form of classical mechanics. Even if we don't go into the cultural arrogance of projecting Eurocentric classical mechanics of scientism to other cultures and ways of experiencing. I do claim also, that Relop has good potential of providing more sound and coherent mathematical foundation for QM, starting from at least two-way process of open interval / duration.

[ScottRoberts]

I also don't see anything gained by saying that pi is not a number. How else to mark the halfway point around a circle? I suppose I should ask first if all the irrationals in his scheme are not numbers? What about e, or i? In measurement, "pi meters" is just as precise as "1 meter". And if not talking about measurement, pi and 1 are equally abstract.

My point was to discuss the distinction between Finnish 'numero' and 'luku', which relates very closely with what can be constructed, ie. written down with numerals, and what can't be. In the realm pure math, Norman's core principle is what he calls 'Law of logical honesty': "Don't pretend to be able to do what you can't do'.

Brouwer-Hilbert controversy boils down to constructibility, and (in my interpretation) Norman takes a very strict principle of constructibility as the founding principle of his approach. Transcendentals etc. irrationals can be "traced" in some magical sense (three dots etc. clues for intuitive imagination), and that magic is the area to which Finnish mathematical term 'luku' refers to, as well as the concept of domains in the meaning of domain of whole numbers, domain of rational numbers. Basically anything where the magical 'three dots' shifts from strict numerical constructibility to intuitive imagination.

Seems to me to be an arbitrary distinction. We usually say that the side of a unit square is a nice simple "1", while the diagonal is the irrational "1.414...", but we could just as well say the diagonal is "1" and the side "0.707...". Similarly for the diagonal and circumference of a circle. Myself, I'd like to see a mathematics where the basic numbers are 1, 0, pi, e, and i, given that they can be combined in Euler's' famous equation e^(pi x i) + 1 = 0. Since they are so relatable, it seems silly to me to distinguish them into two types. By the way, is i a numero or a luku?

In Relop language, the following form can be interpreted and investigated as a landscape of pi:

<>< <> ><>
<<> <> <>>
><> <> <><
<>> <> <<>

Each palindromic line is a different form of basic number-antinumber relation (e.g. positive and negative numbers of Integers). "Equator" can be imagined between 2nd and 3rd line. If we count the "heads" <> with or without "tails" <>< and "beaks" <<>, each line has three heads. and some other processes going on.

How am I supposed to know that <>< means "head with beak" and not "more neither-more-nor-less" or just "more-less-more"? Without some construction rules and a lot of semantic interpretation, I can't even begin to interpret it or see it as investigating anything at all.

The foundational crisis has been about constructibility, the dream of a foundation for mathematical language that would be able to consistently or at least coherently describe continuous movement. Applied mathematics of calculus works to some extent in practice, the crisis since Berkeley is the missing link between applied and pure math.

Some mathematicians might see this as a crisis. Most don't give a damn and just enjoy working in Cantor's Paradise.

My view is that there cannot be a self-reflecting computation, or any self-reflecting mathematics. Mathematics is just form, and for reflection one needs formlessness, which cannot be realized mathematically.

We've been through this before. As Gödel, Escher, Bach shows, Gödel-problematics at deepest level reflect the problem of self-reflection.

No, they just show that we can construct well-formed formulas, or pictures, or music that we can interpret as self-referential. But to say that in themselves they are self-reflective ignores the role that consciousness -- which requires formlessness -- plays for any reflection. Ontologically, I see no difference between "2+2=4" and the Gödel number which we interpet as "this statement is unprovable".

Reflection as such is a form,

I disagree

and self-reflection cannot be realized mathematically in the confines of any static model. Same does not apply to dynamic model, and if process of self-reflection can be coherently characterized by relations <, > and =, Relop should be able to do that.

No, once modelled it will be just as static.

Note the dynamic "near-polarity" of almost palindrome polar-relop. Also the different letters a and the e are almost symmetries on the vertical axis. A difference of difference.

Channeling Derrida, I see.

Here we are going to get into the Cleric/Santeri-Lou discussion, which is unlikely to be any more resolved than that. I will simply state that I side with Cleric, that the animistic mindset was a stage we grew out of, as we will grow out of our current stage. To relate to this topic, I would say that the growing out of this stage will be a matter of self-discipline, of learning to will our thinking, and that mathematics -- any mathematics, set-theory based or not -- is good training (as Plotinus held).

Other words for training are askesis, yoga and kung fu. In the social aspect of mathematical askesis, a certain kind of flexibility and rigour of mind is required. Conjectures are tested, and according to testing, rejected, accepted or further qualified. Important part of the practice is to study and comprehend, why axiomatic set theory should be rejected for foundational, constructible purposes. As well and what heuristic purposes it has served.

Mathematical practice is also very goof for thinking temporal relations and assumptions behind various theories. In light of palindromic time of quantum theory etc. etc., temporal character of theories of consciousness in the form State A -> State B -> State C is very limited form of classical mechanics. Even if we don't go into the cultural arrogance of projecting Eurocentric classical mechanics of scientism to other cultures and ways of experiencing. I do claim also, that Relop has good potential of providing more sound and coherent mathematical foundation for QM, starting from at least two-way process of open interval / duration.

That you want to explore the possibility of working out what one can get with Relop syntax, and Wilderberger's program, and see if it is applicable to QM, and so on is all great -- as I said, an interesting exercise. But that you attach a political/social/metaphysical agenda to it is not so great. It adds an impurity to the askesis ideal.

[SanteriSatama]

Seems to me to be an arbitrary distinction.

Constructibility vs. non-constructibility is far from arbitrary distinction for mathematicians like Brouwer, Weyl etc., and in case of Hilbert, his "solution" was totally arbitrary axiomatics, which broke the communicability of mathematics. Here's a very nice article by a member of this forum:
https://philpapers.org/archive/JONTCE.pdf

We usually say that the side of a unit square is a nice simple "1", while the diagonal is the irrational "1.414...", but we could just as well say the diagonal is "1" and the side "0.707...". Similarly for the diagonal and circumference of a circle. Myself, I'd like to see a mathematics where the basic numbers are 1, 0, pi, e, and i, given that they can be combined in Euler's' famous equation e^(pi x i) + 1 = 0. Since they are so relatable, it seems silly to me to distinguish them into two types.

That was also my question to Norman. Any theory of mathematics should be able to express in some way or another the most beautiful equation of mathematics. He kinda agreed. So far, that goal remains very much WIP.

By the way, is i a numero or a luku?

Depends on the context. In Norman's construction, he's able to demonstrate i as a simple matrix/polynumber.

How am I supposed to know that <>< means "head with beak" and not "more neither-more-nor-less" or just "more-less-more"? Without some construction rules and a lot of semantic interpretation, I can't even begin to interpret it or see it as investigating anything at all.

Yup. Casual discussion is at this phase more fertile to the development of Relop, than fully rigorous presentation. Any case, <>< looks to me like a fish with a tail, and <<> like birds head with a beak, so I in my thoughts I often call the interval <> a 'head'. I can't expect intuitive processes to make much sense, but any attempt to communicate makes some progess, how ever little. Only now that I responded to you, I noticed the very obvious thing that each line of the form consists of three intervals/heads, and then something still mysterious, except that the "tails" and "beaks" attached seem to form a some kind of circle.

The relation of "heads" and "tails" became meaningful, when I observed that we can construct with relational operators Stern-Brocot type trees, where the count of "heads" and "tails" of combinatory strings gives the numerical forms of rational numbers generated by Stern-Brocot.

As my own background is translator, and my main orientation is communicability, this form of social process writing seems to suit that long term goal, as well as to intuit more sharply.

Some mathematicians might see this as a crisis. Most don't give a damn and just enjoy working in Cantor's Paradise.

Academic sociology is what it is.

No, once modelled it will be just as static.

Poetry is not static.

Channeling Derrida, I see.

Oh yeah!

The post-structuralist dude's contribution to the texture of philosophy signifies more and more for process philosophy.

That you want to explore the possibility of working out what one can get with Relop syntax, and Wilderberger's program, and see if it is applicable to QM, and so on is all great -- as I said, an interesting exercise. But that you attach a political/social/metaphysical agenda to it is not so great. It adds an impurity to the askesis ideal.

I understand how you might think so. However, the whole process is a continuum of ethical motivation, and to stay coherent for foundational continuity, the purely mathematical and ethical aspects can't be strictly separated. Myself, I enjoy more talking the purely mathematical aspect, and can make a superficial separation depending on with whom I'm talking. And keep on learning to take the degrees of separation-continuity better in attention.

[ScottRoberts]

Constructibility vs. non-constructibility is far from arbitrary distinction for mathematicians like Brouwer, Weyl etc., and in case of Hilbert, his "solution" was totally arbitrary axiomatics, which broke the communicability of mathematics. Here's a very nice article by a member of this forum:
https://philpapers.org/archive/JONTCE.pdf

I agree with Peter (and Weyl and Bergson) that the real line made of a noncountably infinite number of points does not physically exist, and that the "now" is not a point (as I have discussed in my Time essay). So what? They, with other non-physically existent concepts, exist in the imagination of mathematicians and, spiritual-developmentally speaking, that is all that matters. I am fine with axioms being arbitrary, as long as interesting mental constructions can be built on them.

One can construct the square root of two by drawing the diagonal of a unit square. One can construct pi by taking the length of the diameter and see how many of such lengths it takes to measure the circumference. The fact that neither can be expressed as a rational number just means that there are more numbers than the rationals. My slide rule has a marker "pi" (well, the Greek letter) and what is marked there is no more and no less precisely marked as the marker "2". I fail to see why the fact that if one wants to express pi as a string of numerals, one must end with "...", in any way makes pi less of a number.

What I don't get is why one thinks it is better or truer to found mathematics on what is more "intuitive" to human babies. Does one think that the mentality that imagined the mathematics of physical reality had the same cognitive development that we undergo?

[SanteriSatama]

I agree with Peter (and Weyl and Bergson) that the real line made of a noncountably infinite number of points does not physically exist, and that the "now" is not a point (as I have discussed in my Time essay). So what? They, with other non-physically existent concepts, exist in the imagination of mathematicians and, spiritual-developmentally speaking, that is all that matters. I am fine with axioms being arbitrary, as long as interesting mental constructions can be built on them.

Spiritual-developmentally speaking, axiomatic set theories are dishonest. Real numbers don't form a field, they can't do basic arithmetics. They don't form the continuum they pretend, they don't solve Zeno's paradoxes. Spiritual-developmentally speaking, if the Emperor does not have clothes, it is our obligation to say so and stop believing in dishonest authority.

Even though we may not positively know what the truth is, in our soul we can tell when we are lying, being dishonest. And in our hearts we know that we need trust, and trust requires honesty.

I fail to see why the fact that if one wants to express pi as a string of numerals, one must end with "...", in any way makes pi less of a number.

The original name for a mathematical field is 'arithmetic'. By admitting that pi can't do arithmetic, and hence is not a number, is not disrespect towards pi. On the contrary, by admitting that it is not a mere number under the rules of arithmetic, we show due respect towards its wonder.

What I don't get is why one thinks it is better or truer to found mathematics on what is more "intuitive" to human babies. Does one think that the mentality that imagined the mathematics of physical reality had the same cognitive development that we undergo?

If we human babies are not true to ourselves, how can we be true in the eyes of God? Well, maybe there is no such god that would care for such matters. But even a one can think for a possibility that a spirit such as a/the god of number theory is a spirit of evolving gnothi seauton, process in which human children also participate. Why else would the spirit bother my human life with these mathematical obsessions and inspirations, possessions and intuitions?

Do you really believe it's only one way, top to bottom?

[ScottRoberts]

Spiritual-developmentally speaking, axiomatic set theories are dishonest. Real numbers don't form a field, they can't do basic arithmetics.

You must have a strange definition of "arithmetics". I have never had a problem multiplying 2 times pi.

They don't form the continuum they pretend, they don't solve Zeno's paradoxes. Spiritual-developmentally speaking, if the Emperor does not have clothes, it is our obligation to say so and stop believing in dishonest authority.

I say if you want to explore set-based mathematics, that is good. If you want to explore category-based mathematics, that is good. If you want to explore more-less mathematics, that is good. You say the first is bad. Who is being authoritarian here? And who is engaging in guilt-tripping ("dishonest") to bulk up their authority?

Even though we may not positively know what the truth is, in our soul we can tell when we are lying, being dishonest. And in our hearts we know that we need trust, and trust requires honesty.

I have never felt that I was lying or being dishonest while working with set-based mathematics.

If we human babies are not true to ourselves, how can we be true in the eyes of God? Well, maybe there is no such god that would care for such matters. But even a one can think for a possibility that a spirit such as a/the god of number theory is a spirit of evolving gnothi seauton, process in which human children also participate. Why else would the spirit bother my human life with these mathematical obsessions and inspirations, possessions and intuitions?

I don't know why you are obsessing over this, other than you seem to have a political agenda. I think we are inspired to do mathematics (if we are) because in our current stage of spiritual development the main task is to purify our intellect, to detach it from our lower impulses -- and from political agendas. Thereby, we create a vessel that can accept spiritual knowledge that will increase our self-knowledge and our knowledge of the world. I do not expect any mathematics to supply such knowledge, so it doesn't matter what foundation one works with, and why I said they are all good.

Do you really believe it's only one way, top to bottom?

I don't understand what you are asking. What is "top" and "bottom"?

[SanteriSatama]

I have never had a problem multiplying 2 times pi.

Perhaps you can then share the correct answer with us mortals?

As for the approximations mortals get by punching some buttons of their calculators, they have nothing to do with set theory and real numbers. Floating point arithmetic is rational arithmetic.

What about some randomly picked binary ZFC real numbers, can your divine omniscience also tell does the sum of 0,000... + 0,000... start with 1 or 0?