Bernado's Mathematical Universe

[Squidgers]

Were there no substance, then nothing could be explained in terms of anything. Without substance, when we explain something, then we would always need recourse to another thing, and for that thing, another as well, and so on ad infinitum. Further, if we were to stop explaining a thing in terms of another thing at some point without positing a substance, then there would be no sufficient reason for one cut-off point rather than another, hence explanation in this context is either infinite regress or no explanation at all."

In the quote "substance" is explained by other meanings, which can be further looked upon in a dictionary where words are explained by other words. Reading a dictionary, meaning arises from how dictionary entries relate to each other. The above is an attempt to prove "substance" by assumed absurdity of "infinite regress". Potentially infinite regress, however is not a problem, as completeness is not necessitatated nor assumed beyond axiom of reductionism. And with axiom of reductionism, say hello to Gödel's incompleteness theorem.

Only actually infinite regress of "completed infinity" (such as real numbers) is a genuine problem and absurdity, but nothing a priori necessitates such absurdity, as axiom of reductionism is totally arbitrary metaphysical choice.

"Flowing points", what ever that is supposed to mean, substantial or not (and I'm genuinely curious), at least refers to motion. In the standard canon of Western philosophy, as exemplified e.g. by Kant, substance is supposed to be diachronically permanent, temporally immutable. I'm guessing, but the argument of the book seems to fall into the category of mathematical platonism, immutable and eternal Platonia as the "substance"?

Problem with immutable Platonia is that as historical phenomenon, mathematical languages and thinking have been changing and evolving, and there are passionate foundational disputes between various philosophies and perspectives. Empirically, mathematics is not only found, but also created evolutionary process.

Gödel's incompleteness theorem is just a specific case of the more general Halting problem, as Chaitin explains

The Autho deals with completeness and consistency and frames mathematics outside the formalist paradigm, which I've already said a few times now.

"Eidomorphism as a panpsychism is compatible with both immaterialism and physicalism. This may seem like an outright contradiction at worst or a paradox at best, but in actuality, this follows quite logically from the way the properties of the fundamental substance have been outlined. Given that matter is something unexperienced which is finitely divisible, beyond which there is only void, it follows from two directions — both Sprigge’s arguments and from substantial forms — that matter as the fundamental substance does not exist. Hence, the fundamental substance is by definition immaterial.

Yet, if we think of physicalism only in terms of there being extended entities with ‘physical’ properties — that is, predictable, deterministic, having ‘mass’, etc. — then physicalism can be conceived in terms of eidomorphism as well. Indeed, mass, determinism, predictability, and any other ‘physical’ can all be conceived in terms of mathematico-metaphysical points in motion against a continuum of points, whereby the trajectories of these points conform to mathematical laws. The world of eidomorphism is not an airy dreamworld per se, but rather it consists of real things either in extension or being unextended. Space and time are emanations of the Singularity, and all phenomena are emanations of the Singularity and result out of the basic properties of the Singularity.

Thus, we may also call eidomorphism a mathematical monism or even mathematical idealism because of the ideal property of experience being embedded in a mathematical substance. Even in idealism, mathematics is something invariant in our perceptions, and also what is invariant in our phenomenal experience and invariant in reasoning. As we can see, even in idealism mathematics would be what intellectually describes or even maintains the stability of the world and what guarantees the world’s intelligibility in exact terms."

There is a large section on the description and definition of what ontological mathematics is (the author calls it 'eidetic mathematics' later), but here is a small sample from the section Abductive Proof of Complex Numbers as the Natural Mathematics of Ontology

"The following proof is an abductive proof for lack of a strictly deductive one, though the constraining principles in use eliminate any doubt as to why one explanation will be better than another. We begin by invoking the Principle of Mathematical Completeness, and so it follows that whatever mathematics we must consider to be inherent in the structure of the fundamental substance, it must be a mathematics that is invariant in form under any mathematical operation. To clarify what we are asking, we are considering exactly which number system is the one that has the best chance of being ontological. To answer this question, we need to rule out any alternatives, leaving us with the one and only option. Hence, we are performing a proof by elimination. We can be thankful that given our rather constrained requirements, the proof may be performed in a fairly brief manner since few number systems fit our requirements.

The algebraic analogue to the Principle of Mathematical Completeness is algebraic closure, whereby a field of numbers is closed under all defined operations. A proof of algebraic closure is unnecessary here, but even those slightly familiar with mathematical literature need only look to the Fundamental Theorem of Algebra to understand why the complex numbers form an algebraically closed field.

Our reasoning proceeds thus: no mathematical operation is privileged over another, hence all mathematical operations are ontologically allowed. As inherent forms describing the contingent combinations of modes, mathematical operations are ontological operations, and so the number system inherent to ontology must be able to survive all possible operations. The positive numbers are immediately excluded because they are not closed under subtraction; the integers are also excluded because they aren’t closed under division (excluding zero); the rational numbers are excluded because they cannot take into account algebraic and transcendental numbers (hence cannot perform certain algebraic operations and infinite series); and the real numbers are also excluded because they are not closed under the roots of negative numbers. However, hypercomplex numbers, quaternions, octonions, sedenions, etc. are all types of numbers that cannot perform all algebraic operations, even though they are generalizations of real and complex numbers as well. The sweet spot rests with the complex numbers, namely the projective complex plane, since the projective complex plane also allows for division by zero because there is no sufficient reason to exclude any number from our reasoning.

An additional a posteriori argument for the complex numbers as being the fundamental number system in ontology is seen by simply considering the explanatory success of complex numbers in the fundamental sciences. Complex spaces and the complex projective plane are indispensable to quantum mechanics and all its subsets, such as quantum electrodynamics and chromodynamics. The complex numbers are also found in gauge theory, the unitary groups of the Standard Model, and also the quantum wavefunction. The assurance of their ontological basis is to be found simply in the ubiquity and indispensability of the complex numbers in the fundamental sciences and all of the derivative special sciences as well."

And a section describing how it differs from platonism called The Problem With Platonism and Realism.

"Platonism and realism about mathematics both admit that mathematics is a real, mind-independent structure. The difference between the two is that platonism admits a transcendent and abstract realm independent of matter, space, and time, while mathematical realism is a more general position admitting to the reality of mathematical objects either grounded in matter or some other substance. With realism, there may be a number of different positions regarding what is ontological, and either some or all of our human-derived mathematics is a direct consequence of whatever is ontological. Technically speaking, eidomorphism is a type of mathematical realism that is an expression of an immaterialist but non-platonist neutral monism. For mathematical platonism, there is the dualism of abstract and concrete, and the inherent problem of how the two ever interact. For realism, the problem is that no exact definitions are given for ‘structure’, for matter or a neutral substance, and also the alternatives have not been ruled out. Eidomorphism deliberately avoids grounding mathematics either in pure abstraction or some poorly-defined ontological structure inherent to matter, given the problems with both positions.

Philosophy’s problems with mathematical realism are overdetermined in that there are too many positions which say that mathematics is real, while platonism is underdetermined because it doesn’t account for all phenomena and indeed cannot even account for the concrete-abstract distinction itself. When we talk of realism, if what we consider ‘real’ is something that is independent of our own subjective preferences and what can be accessible to multiple people using an agreed-upon methodology, then we realize that ‘real’ is indeed overdetermined and multiple metaphysical theories will say that certain entities are real, even though such entities may be incommensurable when we compare them to each other. It is at this impasse that philosophy and science have floundered and thus rejected metaphysics. However, if we are asking what is logically necessary and possible, and if we rule out even vanishingly small possibilities that generate absurdities, then we are on the metaphysically correct track to talk of explanatory necessity and ontological necessity.

With eidetic mathematics, when we think in terms of numbers and relate to ontology in terms of numbers, then we are directly experiencing the noumenon and we can directly understand all of existence. When we reason mathematically, there is no reality that becomes mediated by manmade language. Eidetic mathematics grounds numbers on relationships between flowing points, and these relationships may define numbers in terms of ratios of frequencies to frequencies, amplitudes to amplitudes, or even clusters of flowing points to each other. There is, for instance, the open question as to what is the natural base system for mathematical ontology. Thus far, there is no a priori proof of what exactly is the natural, though it’s an open possibility, and we’ll refer to it from here on as ‘base-X’. There is no ambiguity in eidetic mathematics: numbers are concrete relationships between flowing points, whose structure is inherent in an explanatorily necessary, self-perpetuating, self-contained, self-consistent, and analytic ontology."

[SanteriSatama]

The algebraic analogue to the Principle of Mathematical Completeness is algebraic closure, whereby a field of numbers is closed under all defined operations. A proof of algebraic closure is unnecessary here, but even those slightly familiar with mathematical literature need only look to the Fundamental Theorem of Algebra to understand why the complex numbers form an algebraically closed field.

Our reasoning proceeds thus: no mathematical operation is privileged over another, hence all mathematical operations are ontologically allowed. As inherent forms describing the contingent combinations of modes, mathematical operations are ontological operations, and so the number system inherent to ontology must be able to survive all possible operations. The positive numbers are immediately excluded because they are not closed under subtraction; the integers are also excluded because they aren’t closed under division (excluding zero); the rational numbers are excluded because they cannot take into account algebraic and transcendental numbers (hence cannot perform certain algebraic operations and infinite series); and the real numbers are also excluded because they are not closed under the roots of negative numbers. However, hypercomplex numbers, quaternions, octonions, sedenions, etc. are all types of numbers that cannot perform all algebraic operations, even though they are generalizations of real and complex numbers as well. The sweet spot rests with the complex numbers, namely the projective complex plane, since the projective complex plane also allows for division by zero because there is no sufficient reason to exclude any number from our reasoning.

An additional a posteriori argument for the complex numbers as being the fundamental number system in ontology is seen by simply considering the explanatory success of complex numbers in the fundamental sciences. Complex spaces and the complex projective plane are indispensable to quantum mechanics and all its subsets, such as quantum electrodynamics and chromodynamics. The complex numbers are also found in gauge theory, the unitary groups of the Standard Model, and also the quantum wavefunction. The assurance of their ontological basis is to be found simply in the ubiquity and indispensability of the complex numbers in the fundamental sciences and all of the derivative special sciences as well."

Chaitin's paper gives good account why real numbers, and hence complex real numbers, are non-computable.

Non computable numbers can't do basic or any arithmetics, they can't form a field. No matter how many axioms claim so.

You say that the author rejects formalism, but how does he then establish real numbers and FTA? Even wiki admits:

Despite its name, there is no purely algebraic proof of the theorem, since any proof must use some form of the analytic completeness of the real numbers, which is not an algebraic concept.

Fields of real numbers and complex numbers don't exist. Hence they can't form any substance.

[Squidgers]

The algebraic analogue to the Principle of Mathematical Completeness is algebraic closure, whereby a field of numbers is closed under all defined operations. A proof of algebraic closure is unnecessary here, but even those slightly familiar with mathematical literature need only look to the Fundamental Theorem of Algebra to understand why the complex numbers form an algebraically closed field.

Our reasoning proceeds thus: no mathematical operation is privileged over another, hence all mathematical operations are ontologically allowed. As inherent forms describing the contingent combinations of modes, mathematical operations are ontological operations, and so the number system inherent to ontology must be able to survive all possible operations. The positive numbers are immediately excluded because they are not closed under subtraction; the integers are also excluded because they aren’t closed under division (excluding zero); the rational numbers are excluded because they cannot take into account algebraic and transcendental numbers (hence cannot perform certain algebraic operations and infinite series); and the real numbers are also excluded because they are not closed under the roots of negative numbers. However, hypercomplex numbers, quaternions, octonions, sedenions, etc. are all types of numbers that cannot perform all algebraic operations, even though they are generalizations of real and complex numbers as well. The sweet spot rests with the complex numbers, namely the projective complex plane, since the projective complex plane also allows for division by zero because there is no sufficient reason to exclude any number from our reasoning.

An additional a posteriori argument for the complex numbers as being the fundamental number system in ontology is seen by simply considering the explanatory success of complex numbers in the fundamental sciences. Complex spaces and the complex projective plane are indispensable to quantum mechanics and all its subsets, such as quantum electrodynamics and chromodynamics. The complex numbers are also found in gauge theory, the unitary groups of the Standard Model, and also the quantum wavefunction. The assurance of their ontological basis is to be found simply in the ubiquity and indispensability of the complex numbers in the fundamental sciences and all of the derivative special sciences as well."

Chaitin's paper gives good account why real numbers, and hence complex real numbers, are non-computable.

Non computable numbers can't do basic or any arithmetics, they can't form a field. No matter how many axioms claim so.

You say that the author rejects formalism, but how does he then establish real numbers and FTA? Even wiki admits:

Despite its name, there is no purely algebraic proof of the theorem, since any proof must use some form of the analytic completeness of the real numbers, which is not an algebraic concept.

Fields of real numbers and complex numbers don't exist. Hence they can't form any substance.

Here's the section in the book on Chaitin's incompleteness theorem (although it may be easier to read the whole chapter to actually understand the model instead of these out of context snippets):

"Since formal axiomatic systems deal with strings of symbols and their manipulation according to carefully specified rules, they display a phenomenon known as Kolmogorov complexity. Kolmogorov complexity is the measure of how much one can compress some string of information into its most basic building blocks, and of how easily such a string of information may be reconstructed from some given algorithm.

Gregory Chaitin’s own incompleteness theorem deals with statements expressible in any formal axiomatic system, and teases out a result regarding the limits of expressibility of formal axiomatic systems. Chaitin’s incompleteness theorem says that for any formal axiomatic system and for a measure of the Kolmogorov complexity of statements generated by such systems, there exist statements whose measure of Kolmogorov complexity cannot be prov-en to be above some given number. In other words, there is a limit to the statements that a formal system may check for because such statements are far more complex than the axioms of the formal system can account for.

When it comes to mathematics, Chaitin’s result says that mathematics — particularly analysis and number theory — is far more informationally and computationally complex than what is expressible in terms of formal axiomatic systems. This result means that formal axiomatic systems used to describe or even provide a foundational basis for mathematics are incapable of doing so, because the complexity of analysis and number theory exceeds whatever can be captured by formal systems. This immediately puts formal systems on shaky ground, and raises foundational questions for formalist theories like Zermelo-Fraenkel set theory, Peano arithmetic, and Robinson arithmetic. Of course, these formal systems may be enough to describe the entirety of all human mathematical work from now until the end of time, they will still only ever be approximate systems which never capture mathematics in its entirety.

One perfect example of this phenomenon involves the system of real numbers, and the fact that they comprise a nondenumerable or ‘uncountable’ infinity. By extension, the complex numbers also display the same properties, though using the reals as an example will suffice. In his book Meta Math!, Gregory Chaitin gives an analogue of Richard’s paradox, stating that the reals themselves are unnameable. This is to say that there will always be more real numbers than names we can assign to the real numbers, and even if we had infinite time and infinite names, we would still have more real numbers than names. This is a result of Cantor’s diagonal argument which produced the astonishing result about different sizes of infinity.

To make matters more interesting, Chaitin writes in one of his papers that, ‘In other words, any countable set of reals has measure zero and is a so-called null set, i.e., has zero probability and is an infinitesimal subset of the set of all reals.’ What Chaitin is saying here is that any countable or denumerably infinite set of numbers — such as the integers — will always have a measure of zero within the set of the real numbers, or even the complex numbers for that matter. This means that all algebraic numbers are also going to have a measure of zero, as will rational numbers, integers, and any algebraic irrational numbers. The chief composition of the continuum of real numbers consists of transcendental numbers, or numbers which cannot be expressed algebraically, but are often only described by infinite series. The consequence of this means that the bulk of real numbers will never be exactly computed, but only ever approximated.

To illustrate this phenomenon, on the following page is an image of the algebraic numbers on the complex plane. The algebraic numbers are in black, and the transcendental numbers fill up the white space. This is a nice, visual demonstration of the amount of algebraic numbers compared to real numbers. While we can specify many algebraic numbers, there are unimaginably more numbers which cannot be specified. Such numbers fill the white (black in this image) spaces.



The consequence of this result is that any system that uses the real numbers (or, more generally, any uncountable set) will inevitably be far more complex than what can be expressed by finite methods. Even by this simple example, one can see the need for higher-order logical reasoning in analysis, topology, and number theory given that a proper definition of these types of numbers requires quantificational thinking in sets of sets, and so on.

This problem also rears its head in the different types of solutions to equations. Consider, for instance, solutions to differential equations either ordinary or partial. Two main solution types involve analytic or exact solutions, and numerical or approximate solutions. Analytic solutions to differential equations are solutions which are obtained exactly in terms of some solution set and its symbols. Numerical solutions are obtained by running an algorithm to find in which areas and under what conditions does an equation’s solutions come to lie in. However, analytic solutions may either be too difficult to find and be beyond our reach, or even if found, they may be too complicated and time-intensive to give a result in a reasonable period of time. Thus, even with some analytic solutions, a numerical approximation may be desirable.

The difference between analytic and numerical solutions should be obvious given the fact that the number system of choice of differential equations is often the real or complex numbers, or some mix thereof. Continuous quantities and results based on them may have an exact symbolic description and reference, but that symbolic reference might not be computable in finite time. If an exact symbolic description is to be had only in terms of infinite series or indefinite integrals, or integrals over a set (especially a continuous set) then wanting a discrete set of numbers may be too computationally intensive and hence require a discrete algorithm to find some approximation. If differential equations are cast over such number systems, then it’s only a matter of time before mathematicians and physicists begin to discover ones whose solution sets are not easily or even finitely computable.

This type of problem has direct consequences for computability theory and Turing completeness when it comes to using algorithms to deal with number theory and analysis in particular. As Chaitin writes in his paper on unnameable reals,

‘The set of all possible computer programs is countable, therefore the set of all computable reals is countable, and diagonalizing over the computable reals immediately yields an uncomputable real. ... In a nutshell, the set of computer programs is countable, therefore the set of all computable reals is countable, and therefore, as in Sec. 2.2, of measure zero. ... What if we allow arbitrary, highly nonconstructive means to specify particular reals, not just computer programs? The argument of Sec. 2.5 carries over immediately within our new framework in which we consider formal languages instead of natural languages. Most reals remain unnameable, with probability one.’

By a simple syllogism, it follows that the set of all possible programs cannot compute the set of all real numbers. Given the fact that there are also uncountably many theorems that may be made with regards to the real numbers (and by extension the complex numbers), and also from the fact that there are uncountably many mathematical structures over the reals, it also follows that the total number of theorems must also be uncountable. Thus, when there are yes or no questions regarding real or complex analysis that deal with a mathematical structure inaccessible from any number of formal axiomatic systems, it will follow immediately that such questions are formally undecidable, as there will always be more continuous entities than discrete algorithms to compute them.

Eidomorphism defines everything according to infinitesimal points whose cardinality is consequently uncountable, and hence formal systems will never be able to capture the truth of eidomorphism by axioms alone. Eidomorphism also implies that there are only analytic solutions to differential equations which pertain to ontology, and numerical ones are an approximation and will always remain so. The correctness of the approximation will always be determined not by the numerical method or its analogical comparison to some real phenomenon, but by deductive reasoning in analysis itself. This is especially pertinent if a differential equation has an exact solution and the proof of the solution is of a complexity that far exceeds anything that can be captured by a Universal Turing Machine.

Bekenstein Bound
Chaitin also mentioned that reality may be discrete, and quoted Smolin in describing the Bekenstein bound of information with regards to the maximum amount of information in a finite region of space13 . In Chapter 7, we formally define flowing points and also their lower bound. From this, Eidomorphism provides an answer to the Bekenstein bound and conceptually resolves the paradox between the continuous and the discrete. Since we define space by infinitesimals, it follows that the Bekenstein bound of the fundamental substance is effectively infinite. However, because flowing points have a minimal bound with regards to their motion, the Bekenstein bound of any enduring structures composed of flowing points will have a Bekenstein bound similar to the one measured in physics thus far. So, while spatial structures may be discrete, they all exist against a continuous background structure determining the nature and laws of the spatial structures. Thus, to understand the universe entirely, a computable mechanism can only produce an approximation, though an exact description or exact knowledge of the continuous structure may be had by analytic or symbolic means. Mathematical analysis and number theory are themselves the standalone methods of gaining exact knowledge of the fundamental substance, and only these methods coupled to eidetic mathematics can lead to an ontologically and mathematically complete and consistent approach to understanding mathematics."

[SanteriSatama]

Eidomorphism defines everything according to infinitesimal points whose cardinality is consequently uncountable, and hence formal systems will never be able to capture the truth of eidomorphism by axioms alone. Eidomorphism also implies that there are only analytic solutions to differential equations which pertain to ontology, and numerical ones are an approximation and will always remain so. The correctness of the approximation will always be determined not by the numerical method or its analogical comparison to some real phenomenon, but by deductive reasoning in analysis itself. This is especially pertinent if a differential equation has an exact solution and the proof of the solution is of a complexity that far exceeds anything that can be captured by a Universal Turing Machine.

Very well, at least non-computability is acknowledged. Eidomorphism is founded on arbitrary axioms of reductionism and independent existence of points. The essence of formalism is arbitrary axiomatics.

Deductive analytical methods (Zeno's paradoxes) tell that motion is not possible in point-reductionistic ontology. In analytical context of actual infinity (which is self-contradictory as such) the notion of 'infinitesimal' cannot be any discrete value, it's by definition an infinite regress - which the author considered an absurdity elsewhere. I don't think honest deduction would allow such incoherence?

Hence the notions of "flowing" and "point" are mutually contradictory. With the intuitive demonstration of point as end of line, as defined by Euclid's 3rd definition, the point is a discontinuity and I can't see any coherent way to derive continuity from such discontinuity.

Postulating non-computational things like "real numbers" "complex numbers" and "complex plane" from point-reductionist ontology and declaring that they can do basic arithmetics (ie. form a field as rational numbers do by mathematical induction) cannot be a valid deduction, it's an arbitrary and absurd make-believe axiom.

Lebesque measure (e.g.: "Any closed interval [a, b] of real numbers is Lebesgue-measurable, and its Lebesgue measure is the length b − a") offers no help, as the likelyhood that a and b are non-computable pseudo-transcendentals is 100%, and hence "b - a" has no meaning.

Further, how can Archimedean property be derived from point-reductionist ontology, how can any measure of infinite set of points be more or less than any other measure? Point-reductionism does not allow coherent derivation and deduction of most foundational relation of mathematics, the more-less relation.

You see the problems of incoherent and seeminly illogical deduction in the reductionist's argumentation? Has the reductionist addressed these problems to your satisfaction?

***

PS: The situation changes radically, if we take undecidablity of Halting as foundational, and qualify infinitesimal and infinite as potentially infinite processes with potential halting. As Chaitin shows, incompleteness is special case of Halting, so it's also included in the foundation.

Potentially, but not actually infinite processes/measurements/computations of 'both more and less' and 'neither more nore less' are perfectly meaningful relations without actual infinite regress.

The substance of point-reductionism is the maximal violation of the principal of parsimony 'entia non sunt multiplicenda'. Relational process ontology constructed from relational operators < and > does not postulate any 'entia', and is thus maximally parsimonous, with coherent ability to deduce and explain also discrete phenomena.

[Ben Iscatus]

Santeri, have you ever considered that a computed reality could cause the incommensurabilities and paradoxes of mathematics (such as flowing vs point and quantum theory vs Einstein's relativity)? Have you dismissed the idea?

[SanteriSatama]

Santeri, have you ever considered that a computed reality could cause the incommensurabilities and paradoxes of mathematics (such as flowing vs point and quantum theory vs Einstein's relativity)? Have you dismissed the idea?

It's very clear that there are incommeasurabilities between various metrics, and computational recursiveness seems to hold only for relations of repetition and nesting. So, there are many issues that produce computational irreducibility aka undecidability, at least the discrete domain, as well as local irreducibilities (e.g. computable equilateral triangle does not exist in 2D, but does so in 3D).

The combinators that Wolfram has been studying as the basis of his hypothesis and approach to computational universe are very interesting because they are very 'abstract' in the sense that they are metric-independent and thus very general. Relating various metrics and geometric phenomena of incommeasurabilities is a huge and very complicated practical challenge, but not impossible in principle.

Wolfram's approach does not as such provide coherent modelling and ontology for computation durations. My hypothesis that mereology of Bergson durations could provide such is not exactly a "metric", in the sense that duration is neither unity nor multiplicity, it's defined as indefinite undecidablity. With construction of duration as 'both more and less', as open interval consisting relational operators <>, durations can be mereologically related.

Such foundation expands mathematical and computable qualia beyond discrete computation to large variety of qualia which can be expressed as comparative more-less relations. But I'm not suggesting that all qualia can be reduced to comparative relations, or any ontological reduction of qualia to computation. However, correlations between qualia and computation is as such very fascinating phenomenon.

[Squidgers]

Eidomorphism is founded on arbitrary axioms of reductionism and independent existence of points. The essence of formalism is arbitrary axiomatics.

No, you still have no idea what Eidomorphism is founded on. You need to drop your assumed framing of the model and understand the actual framing of the model. Actually reading the book would help.

You are arguing against a strawman.

You keep saying nonsense like this then writing paragraphs to show why your own made up false assumptions are wrong.... so I'm not sure if you are getting it.

Here's yet another example, hopefully clearer this time:

Bob finds a new idea that doesn't fit into the usually prescribed categories of capitalism or communism. He shares that model briefly with Sam. Sam dismisses the model saying it is just communism. Bob says it isn't, and that there are many nuances in the writing that show why. Sam refuses to actually read his model, instead preffering to assume it is communism and argue for hours why he thinks communism is bad.

Bob tries to explain that Sam is fighting a strawman, not the ideas he is putting forward. Sam gets confused and asks why Bob is making the conversation personal instead of joining his discussion about communism.

Sam continues to argue his criticisms of communism whenever Bob brings up his model.

Bob writes an overly simplistic story about the situation to try and explain what Sam is doing.

[SanteriSatama]

You are arguing against a strawman.

No, I want to talk math, and see whether we - you and me - can reach consensus on cruxial issues. I don't give a rats ass about your authoritarian book worship and religiouss peddling. Please address the math issues I raised in the previous post, according to your own best comprehension, or shut up and go away.

[Squidgers]

You are arguing against a strawman.

No, I want to talk math, and see whether we - you and me - can reach consensus on cruxial issues. I don't give a rats ass about your authoritarian book worship and religiouss peddling. Please address the math issues I raised in the previous post, according to your own best comprehension, or shut up and go away.

We will never reach consensus while you refuse to explore the necessary meta conversations first.

Here it is one more time.

You are arguing from a particular mathematical position. im trying to tell you that your position is invalid when discussing the concepts in the book because it is offering a different position to interpret mathematics from.

So it's not a mathematical conversation, but a philosophical one that is needed before any communication can happen.

What paradigm of mathematics are you coming from? Why choose that one over another? What assumptions about mathematics does your paradigm have? How do these assumptions lead you to your conclusions, which might otherwise be different under different assumptions?

[SanteriSatama]

What paradigm of mathematics are you coming from?

Paradigm of coherence theory of truth.

Why choose that one over another?

Coherence theory of truth is primary to pragmatic theory of truth, as ethical and coherent action comes from holistic comprehension. Foundationally and ontologically, there are no "different mathematics", which would lead to fragmentation and loss of communicability. Foundational coherence is a process and purpose, which may not never be fully reached and completed.

What assumptions about mathematics does your paradigm have?

Coherence theory of truth does not reduce to if-then -type heuristic structure of formalism. Coherent mathematics is not based on assumptions in that sense. We strive to make coherence-conditions as inclusive and general as we can, but they can't be fully formalized.

The primary values of mathematics and constructing mathematical languages are truth, beauty and communicability. Empirism, intuition and phenomenal mathematical cognition cannot be excluded from conditions of coherence.

How do these assumptions lead you to your conclusions, which might otherwise be different under different assumptions?

As said, coherent conclusions are not derived from arbitrary assumptions.