A Manifesto

A Manifesto

There is exactly one absolute: There is no such thing as a world without boundaries nor a perfectly impermeable boundary <=> the only absolute is there are no absolutes <=> there are no ‘perfect’ boundaries, there are only filters

• The continuum defines all boundaries, not the other way around…echoes of Deluze and Derrida

• Where is the perfectly impermeable boundary?….show me (Wittgenstein)…let me touch it, hear it, smell it, taste it.

• ‘Infinity’ is just another word for ‘indeterminate’…”yet to be determined”

• ‘Being’ has no meaning in a world where ‘the infinite’ is ‘determined

• Claiming that the continuum is “uncountable” is the same as claiming that there exists an interval that cannot be divided into four pieces…nonsense.

• ‘free’ vs ‘determined’ can never be determined, therefore ‘free’ “wins”….’X’ always has a winning strategy in a game of tic-tac-toe on a board of fixed extent (horizontal/vertical dimension) when there is no limit on the number of degrees of freedom (‘vertical’ dimensions)…Hales-Jewett theorem.

• The idea that all boundaries are permeable is the idea of intuition…that Nature is both structured and ‘unstructured’….and cannot be modeled accurately by fixed symbolic manipulation without a reference to or allowance for an additional unknowable/inarticulable degree of freedom….”Whereof one cannot speak, thereof one must be silent”(Wittgenstein)

• There is an arrogance, a chauvinism in our sense of ‘self’…clearly we are intimately connected to ‘the outer world’. An organism is more than ‘the sum of its parts’ mainly because the drawing of any boundary is fundamentally arbitrary, at best pragmatic, operational. ‘Pragmatic’ implies a goal, and all goals are themselves arbitrary, teleological, and often founded in a kind of chauvinism…’me’ or ‘us’.

• The distinction between what can be counted and the continuum is absolute (all boundaries are porous) therefore, what constitutes ‘things’ to be counted is not…therein lies permeability, an instability of boundary.

• There is no such ‘thing’ as a perpetual motion machine…everything is perpetual motion …all boundaries are permeable

• Continuity is both balance and imbalance…there is no ‘point’ of perfect balance… there is no ‘actual infinity’… the ‘point’ of reference (about which the universe ‘turns’…and the universe is observed to be ‘turning’) is off-center, but only by appearances…. everywhere there is a net ‘torque’. An apt metaphor is ‘the scale of justice’. A ‘measurement’ consists in placing an ‘object’ in the left pan and then placing decreasing weights (starting with 2^n and decreasing in powers of 1/2) in the right or left pan according to which side ‘falls’…we never place and remove a weight…we just add a weight to the side that rises….the process never actually ends until someone says, “enough.”

• Turn Sharkovskii’s proof on its head… the obvious ‘imbalance’ of a cycle of period three as opposed to the obvious ‘balance’ of a cycle of period one (the two of them are clearly ‘opposed’) is a clear indication that the ITV (Intermediate Value Theorem) is nonsense…simply a consequence of arbitrarily denying any additional degree of freedom.

• Huygens’ wavelet model is an apt model of the universe…every ‘point’ is potentially both source and/or sink, information clearly is not uniformly distributed. The distinction between ’program’ and ‘data’ is arbitrary, linguistic, teleological.

• It is a fact that there is some order to the universe (albeit transient, time dependent) or else we wouldn’t see a linear track in our cloud chamber.

• Bell’s Theorem requires us to acknowledge time

• Sierpinski constructed his triangle via the iteration of a pattern of connected line segments arranged on a hexagonal grid … an infinite (never ending) process… a ‘gasket’ filling curve. At the n’th iteration each line segment can be addressed in sequence in a ternary Gray code that exhausts the set of possible ternary sequences of length n…not possible in binary…counting in binary on an object with a threefold symmetry fails to ‘map’ the symmetry…literally, the map is not the territory

• It has recently been found that the ‘infinite’ plane can be tiled non-periodically with a single shape based on a hexagon….yet there are ‘islands’ of order even so.

• In some way everything ‘leaks’… the best we can do is find leaks. There is no way to ‘stop’ all the leaks.

• We Europeans are so enamored of our ability to exploit boundaries in the form of pistons and valves (and their electronic and social equivalents) to make things ‘work’ that we fail to appreciate the extent to which such boundaries ’leak’ (‘unintended consequences’, R Merton)…climate denialism, blindness to the loss of biodiversity, denial of debt owed to those we killed or enslaved, denial of death.

• If we live like there’s no tomorrow, then there will be no tomorrow…to that extent the fantasy is real.

• The whole reference to death as “giving up the ghost” points to the depth in our culture of the idea of boundaries in the form of ‘possession’…and that idea’s internal contradiction: in as much as we have no control over death, the idea that we ‘possess’ our bodies is absurd….and if not our bodies, then what sense does it make to think of ourselves as ‘possessing’ anything?

• Killing and slavery are expressions of the will to possess, to ‘own’ others as a denial of our inability to own ‘ourselves’. (Becker, ‘Escape From Evil’ – Graeber, ‘Debt’)

• Our bodies are the ultimate example of a boundary with holes….pores everywhere. When is the oxygen we inhale part of ‘us’…or the carbon dioxide we exhale not part of ‘us’…or the water and food we take in and excrete, the sweat we exude? At the boundary of our cells? Our cells die and are replaced constantly…and so with us. How absurd it is that we who are surrounded by permeable boundaries, whose whole existence depends on permeable boundaries, would assume the existence of a perfectly impermeable boundary….Platonic nonsense.

• Given that social behaviors can and do influence mating choices, and mating choices influence genetic inheritance, there is no way to draw a definite boundary between our biology and our sociology. In that sense, our social behavior is a ‘self-fulfilling prophecy’….a self-referential/recursive process.

• After WWI, the European deceit of setting arbitrary boundaries and possessing ‘spheres of influence’ exacerbated the intractable problem we see today in the Middle East. Religious extremism and secular hubris abide no boundaries.

• The most fundamental structure is a kind of duality: complementarity (Elitzer-Vaidman bomb tester)… reality is flux…in that Heraclitus was correct.

• There is always at least one ‘degree of freedom’…for every ‘event’ there is at least one ‘non-event’…nothing is absolutely determined.

• The universe clearly is simultaneously ordered and disordered…there is always an ‘escape’, though not in the form ‘we’ would ‘like’.

• Order is both local and universal, disorder is by definition the permeability of all boundaries…the fundamental error is assuming a perfectly impermeable boundary

• What ‘we’ call an ‘object’ is a relation (what we see as an object is a consequence of our evolution…we relate to ‘it’ as an object)

• It is possible that there is no choice function for pairs in a ‘set theory with atoms’, Fraenkel(1922)

• Consider Freiling’s Axiom of Symmetry and the construction of the Vitali sets.

• A binary string is an element in an indicator function over N. The elements of N are the atoms of a set theory. As Fraenkel showed, it is consistent with such a set theory that there can be no choice function for disjoint pairs of atoms…all infinite binary strings are indeterminate.

• Much of what we call ‘real’ analysis is actually a science of approximation…the ‘realist’ is fantasizing…their fantasies are real only to the extent that fantasies can have real world consequences (consider the history of wars of religion and ‘culture wars’).

• A ‘test function’ is nothing more than a step of 1 on the ‘line at infinity’ in projective space …the sole basis for an equivalence between area and length….the link between magnitudes in two otherwise distinctly different categories

…what is calculus really?… a link between two distinctly different categories: line segments and areas…distributions simply bury the link one step ‘deeper’….yet there is no escape from the fundamental step of ‘approximation’ that is the essence of all analysis. The Archimedean Axiom amounts to a ‘rounding off’ of an infinite regress…an arbitrary cut off determined by ‘good enough’ or ‘good as possible’….’good’ is determined by ‘the goal’, and goals by beliefs, by culture, by society.

• There is no absolute…there is only what we do.

• Mathematical ’rigor’ amounts to arbitrary assumption plus valid reasoning, not Truth. The assumptions are motivated by a goal orientation. That orientation is determined by belief.

• There is no ‘outside’ = there is no ‘empty set’, no ‘void’, no absolute ’zero’ …there is only a lack of knowledge

• All binary opposites meet at a permeable boundary…everything is connected

• With Weierstrass, “we will banish time…” mathematicians became advocates for determinism. However, time is real, the 2nd ‘Law’ of Thermodynamics is reasonable. The ‘spacialization’ of time is nonsense (R Unger). Non-local quantum objects ‘exist’.

• The idea of a completed infinity is nonsense. It is the logical outcome of “banishing time”

• We see objects in our ‘macro world’…it is a matter of the scale of our existence. The role of chance becomes clear when we examine our world at the scale of the quantum. Our world truly is not ‘punctuate’, though we describe it as such. As we are limited to the finite, it is a fundamental error to carry our ‘macro world’ framing into the micro world of the quantum on the back of a fantasy world of the completed infinity…a Bayesian view is the only one that makes sense…E T Jaynes was right…while the real world ‘exists’, our only tool for its ‘apprehension’ is probability.

• Process (the continuous) is real…we experience it as time…the discrete (the static) is theoretical (an abstraction). Because Parmenides and Zeno thought of reality in terms of a duality, the continuous and the static, they were only half right: ‘Infinite subdivision leading to absolute zero’ is a fallacy. There is no absolute zero…No outside, no void, no perfectly isolated system, no fixed state…there is only everything. The only thing fixed about ‘it’ is our representation of it.

• finite/infinite: a binary concept within Aristotelian logic…a two state logic. A ‘completed infinity’ introduces a third state, a third ‘truth value’ and hence is outside Aristotelian logic and cannot be represented using a binary code (without a prestidigitation). Truly, a ‘completed infinity’ is an infinite regress as well as an instance of circular logic (‘Naming’, Gaifman: “…two sides of the same coin”), thus cannot ‘fit’ within a two-state system of logic.

• “The main challenge in a precise statement of Stokes’ theorem is in defining the notion of a boundary.” The notion of ‘boundary’ is dependent on the idea of ‘the continuum’.

• Kronos and kairos, number and magnitude…the ancient Greeks already were aware of the boundaries of logic. You cannot count objects until you assume boundaries on things to be counted….yet all boundaries are permeable…there are no Platonic (‘perfect’) boundaries. Plato was a fascist.

• The continuum is non-Archimedean

• The correct logic for a full appreciation of our world is an intuitionist logic. It is not a three state logic. Rather, it is a two state logic with the possibility that the state of a proposition is indeterminate (yet to be determined).

• Time is real. Smooth Infinitesimal Analysis provides a mathematical model. Dual numbers via 2 x 2 matrices is its simplest expression….a geometric algebra.

• ‘determinism’ vs ‘free will’ is a binary with a porous boundary…the question is for us unanswerable …the best we can do is address individual events probabilistically…we can approximate.

• Mathematics vs. Biology: Evolutionary theory founded on the historical record clearly shows the arrow of time. We are animals. We are a message traveling through time, generation after generation …it’s in our genes. Our awareness of our own existence is evidence of the reality of our heritage. The records studied in biology, geology, paleontology and archeology provide details.

• The evolution of species is clearly a stochastic process. Nature is indifferent to our existence.

• The idea that the universe is deterministic (Sapolsky) would imply that our thoughts are determined, but that would imply that our thoughts about our thoughts being determined would also be determined, etc., leading to an infinite regress = a truth gap (‘Naming’, Gaifman)…thus we have a boundary with holes. (‘Münchhausen trilemma’, H Albert)

• Nature vs nurture will not be answered by biology alone…that boundary is indeterminate. Those making claims to the contrary are simply staking out a claim to a ‘territory’…territorial like a dog… pissing on every lamppost. Sadly, as a people we have not learned.

• Time dilation may be real or at least characterizable as relative, but the ‘arrow of time’ is real (the Mott Problem)

• We are the way we are because we happen to have the history we have…there is no ‘necessity’ to the way we are. (the Mott problem: spherical wave yet linear track).

• Any ‘thing’ other than me has a different unique history…any different linear trajectory in a cloud chamber (the Mott problem) has a unique history

• AI will never produce another ‘me’ simply because it can never have the same history: my ‘history’ is continuous with ‘me’…we are not ‘distinct’ in the sense of separate…I would not exist, if it were not for my history…to separate me from my history is just a “mind game”…it’s unrealistic…time is real.

• Everything that we know has a history and nothing is ultimately ‘determined’… neither by ‘first cause’ nor by ‘final cause’ (‘Causation’, Norton)

• Many mathematicians (along with the religious and the politicians) have turned chance into destiny, have succumbed to the allure of ‘final causes’, to teleological thinking.

• No system of axioms can ‘bound’ mathematics (Skolem, Godel)…the argument for all of set theory as the sole basis for knowing is ultimately circular and self-negating.

• No language can define its own Truth predicate (Tarski). The representation of ‘all’ rational numbers is sufficient to exhaust the representational ‘power’ of decimal notation (or any other finite base notation). The complement (in R) to the set of rationals is not a decimal. The algebraic quotient R/Q+ is topologically a single point. The simple linguistic move of referring to an irrational number as an “infinite decimal” does not and cannot determine an impermeable boundary between the two categories rational/irrational …it is a nonsense expression

• Each coset in R/Q+ [0,1] (essentially r + Q [0,1] for some r in R [0,1]) is an equivalence class of Cauchy sequences whose ‘limit’ is r… except that while each q in Q is of finite length when represented as a continuing fraction, the length of all such fractions, all of Q [0,1], is unlimited (there is no smallest positive rational in [0,1])…therefore there is no way to establish the uniqueness of r via continuing fractions …no non-circular way to distinguish one ‘equivalence class’ from another….R/Q+ [0,1] is a single point. The whole ‘Cauchy construction’ of the ‘Real numbers’ fails.

• For some, mathematics is a means for colonizing minds. Its power is instrumental, not absolute. British rule in India came to an end…Britain still has a king and a Church …both are hollow husks, cultural artifacts. The future is not determined.

• The Stern-Brocot tree is an infinite binary tree…its elements exhaust the rational numbers and each can be given a unique, finite binary representation.

• Each binary entry in the S-B tree corresponds to an algorithm that generates it…as Turing showed in his Halting Problem paper, we know that there can be no algorithm that can determine in general whether a given algorithm will halt (…be resolved in a finite or ‘fixed’ amount of time)… as the row number in the tree increases, the number of algorithms that can generate a particular output increases at least exponentially (it is the ‘algorithms’, the processes that are “uncountable” not their output…process is everything)…’the particular’ becomes ‘the general’, the ‘existential’ becomes the ‘universal’, thus the boundary between the rationals and irrationals (the ‘leaves’ vs ‘branches’ of the tree) must be indeterminable. Attempting to make the distinction via linguistic terms such as ‘nodes’ and ‘paths’ is attempting to make the very same distinction. The word ‘path’ applied in the sense of independence from a node is simply a restatement of Zeno’s claims regarding the static and the dynamic. While both are descriptive of reality, they are not equivalent…there exists an essential imbalance in Nature.

• To the extent that a number is definite it is rational…if a number is not rational then it is not definite (its extension defined as its radix representation). While we can define an irrational algebraically, algorithmically or linguistically (all of which are essentially equivalent…Gaifman, ‘Naming’), we cannot define it via radix notation…it is simply not possible in fact…any such attempt will be self-referential and hence an infinite regress …circular (in projective geometry the line at infinity is a also characterized as a circle of infinite radius, and like the horizon it recedes upon approach)

• While we can know in advance that some algorithms will halt or not, we cannot know in general in advance which algorithms generate a binary string that will correspond to a rational vs irrational number, thus the truth of the statement that there is a definite boundary between the two categories is itself indeterminable. Therefore, we can think of the rationals in general as the output of some algorithm that _happened_ to halt.

• We cannot in general determine if a given string will _happen_ to terminate (…be resolved in a finite or ‘fixed’ amount of time) and be rational or not: the category/name ‘infinite decimal’ is really a reference to that indeterminacy…and that is exactly what an ‘irrational number’ is. The name ‘infinite decimal’ is a misnomer, a red herring, aimed at ‘leaping over’ that chasm (think Kierkegaard) to an imaginary world where there is no arrow of time and we can accomplish what Turing showed us is impossible. Hence the ‘ultimate’ Turing ‘Oracle’…otherwise known as the Axiom of Choice….is truly religious nonsense.

• The Axiom of Choice does not reconcile the random with the determined. The reality is that we cannot ‘make peace’ between them except by arbitrary choice…by ‘choosing’ sides.

• The only sensible position is that of the ‘anti-realist’ (the attitude that makes a democracy possible)…’realists’ are just ‘crypto-Nazis’ (G Vidal vs W F Buckley)

• While we can by definition (linguistically) differentiate between the rational and the irrational, the boundary between them cannot be found (or ‘defined’) via numerical (decimal, binary. etc.) representation. We can at best approach that boundary (while an individual rational will correspond to a terminating string, there is no limit to the length of the strings necessary to represent all individual rational numbers…there is no smallest positive rational number). The idea of ‘a dense set’ does not “solve the problem”. Zeno was half right. There are clearly “two sides to the story”. The expression “infinite decimal” is empty of meaning.

• A fundamental error: attributing the boundedness of the finite to the infinite…the ‘gap’ is irremediable.

• In ‘real life’ complete is a relative term…it is equivalent to ‘good enough’ relative to some socially or personally determined goal. Likewise, in mathematics ‘complete’ is a relative term, relative to some socially or personally determined goal. In both cases it is a matter of culture.

• Those who believe that technology is our best solution to the problems of our existence are just as wrong as those who believe that “God” has determined our future. Technology can and does provide solutions to particular problems at hand, but the ‘real problem’ of our existence is outside of the realm of technology…our ‘real problems’ are behavioral, cultural ….their solution begins with recognizing that there are no absolutes (Becker,’Escape From Evil’). “It is what it is.”(‘The Irishman’, DeNiro)

• Every step we take is completely and utterly arbitrary in its origin, therefore we are free to choose our next step. We are limited to teleological thinking only by belief. There is the impossible, but not everything is determined…there is a ‘gap’.

• In the world of the punctuate: by the Weiner measure the set of all differentiable, continuous functions is of measure zero…that is, almost all continuous functions are non-differentiable…‘pathological’…such functions can be thought of as having no definite direction at every (or even almost every) point. The most ‘pathological’ are indeterminate in form…have no fixed ‘global’ structure.

• Mathematics is a ‘prophesy’ in search of fulfillment (a model) …that structure is clear given proof by contradiction (‘Pessimism’, Cohen) There is no need to search for a model…it’s here: the real world, the world of sensation, the world where time is real.

• Consider the ‘prophesy’ that there ‘exists’ such a thing as an ‘infinite set’….is that a prophesy that can ever be fulfilled in the sense that any other prophesy may or may not be?….in other words: at some time? What is real?

• We cannot reasonably say, “the universe has a boundary”…only our knowledge of it is bounded and that boundary recedes upon approach 

• There is an exception to every rule except the one absolute….every set has a complement except the set of all sets that are members of themselves…its complement is a ‘truth gap’ (‘Naming’, Gaifman: “…two sides of the same coin”)

• What lasts ‘forever’?…we cannot possibly know or even infer, except exactly that we cannot know 

• The finite can never be a measure of the infinite…what cannot be measured is infinite…that for which no unit can be consistently established is infinite.

• The only ‘fundamental unit’ is the recognition that there is no such thing as a perfectly impermeable boundary and that therefore everything is one

• Everything is connected. There is no ‘outside’ …all units other than the ‘fundamental unit’ are arbitrary/contingent …’mind’ can be described at all scales where there is any kind of connection… any ‘sharing’ (panpsychism)

• Features of the distinct unit and features of continuity (simultaneously infinitely larger and smaller ‘units’) are both ‘at work’….how we ‘see it’ depends on what we look for, what we look for depends on what we expect (theory drives experiment)…experiment may reveal something unexpected

• Everything is connected…the discrete only exists in our memory of the past and our memory of the past is bound to be incomplete… the map is not the territory

• The infinite set is a ‘god like’ construction…All ‘god like’ constructions are nonsense…we are not ‘gods’…we can assume a view towards infinity, but we cannot reasonably assume a view from infinity. What else is the idea of an infinite set than a claim to a view from infinity, from the ‘end’ of an infinite process, from a ‘point’ outside. Every ‘outside’ is just another hole in a larger structure. (an infinite Sierpinski gasket)

• Projective geometry and ordered geometry can individually serve as a foundation for all of geometry (Cayley, Pasch), both are realizable in the world of sensation, but not simultaneously …there is no such thing as a ‘closed, totally ordered field’ (except by approximation, but then how ‘closed’ is it?… partial, yes…total, no)…it is a fiction without support in the world of sensation

• The distinction between “ideal”(Hilbert) points and ordinary points is nonsense… the idea of a ‘point’ is ideal in itself. It is a matter of the ‘point of view’, and the one point of view that is impossible to attain is the view from infinity, from the ‘end’ of an infinite process. Furthermore, what you thought was a point upon closer examination turns out to be a segment.

• The Axiom of Choice is nonsense (Banach-Tarski Paradox). There is no such thing as a perfect analogy…that is a contradiction in terms….”the map is not the territory”

• The reason the Banach-Tarski paradox ‘works’ is that the claim that there exists the means or ability to distinguish ‘points’ with absolute precision (the AoC) is false (in particular when expressed as binary strings), hence we end up with a duplication…a ‘Mind Projection Fallacy’ (E T Jaynes). The ‘boundary’ between ‘points’ expressed as binary strings is indeterminable. There is no escape from giving ‘intuition’ a role in mathematics. The AoC produces counterintuitive results…it amounts to wishful thinking…a deliberate ‘forgetting’ about the ‘other half’ of the duplication.

• Probability theory (the logic of science E T Jaynes) is the proper foundation of mathematics, not the other way around

• All quantifications other than the one absolute are at best approximations. Even the count of some conventionality defined, finitely populated category is an approximation. Every category is conventional/contingent and not absolute.

• What is an ‘object’?… ‘things’ in the world do not present themselves to us with labels attached…that we see ‘things’ as objects is a consequence of our evolution (G Edelman)… boundaries exist yet all boundaries are porous.

• “I see three things in the field…two cows and a pair of cows”…(G Ryle)… “Is the bung hole part of the barrel?”

• It is possible that there is a ‘reality’ currently ‘outside’ our world of sensation, outside the concepts we form based on our sensations… such a world would make ‘itself’ known in time since the boundary between that world and our world of sensation must be separated by a permeable boundary (the one absolute). There is no ‘absolute outside’. Until such time all attempts to assert the existence of such a world with no supporting evidence in the world of sensation are by definition nonsense. Until such time the best we can do is apply our naturally endowed abilities augmented by proper training for determining likelihoods: probabilities (Kahneman, Jaynes)

• All models that have no demonstrable support in the world of sensation are by definition not real and hence are religious/political nonsense and simply part of a status ‘game’, a contest, (generally in some way violent)…that game is real only in the sense that the boundary between the game and the real world is permeable <=> the game has real-world consequences (it has the form of a “self-fulfilling prophecy” R Merton)

• Turn Plato on his head….it is the abstractions and symbols (the ‘perfect world’) created by us that are the shadows on the wall…there is only the one real (‘imperfect’) world. We are animals. Aging is real. Death is real.

• Any attempt at ‘a view from outside’ is a fool’s errand or worse, a quest for domination

• Foucault is correct : there is most definitely a nexus between knowledge and power.

• Any model that lacks a stochastic element is nonsense. All mathematical models that ‘fit’ within our naturally endowed abilities to assess or quantify probabilities are ‘valid’ models, all others are nonsense and not mathematical until shown to be otherwise by way of a probabilistic model supported in our world of sensation.

• The boundary between mathematics and nonsense is permeable…what is ‘mathematical’ changes with time (Poincaré 1912)