### Some layman considerations related to the fundamentals of mathematics

I am not a mathematician and therefore should refrain from consideration of anything related to fundamentals of mathematics. In the discussions with Santeri Satama I could not avoid the temptation to break this rule. I however feel that I must confess my sins and in the following I will do this.

- Gödel's problematics is shown to have a topological analog in real topology, which however disappears in p-adic topology which raises the question whether the replacement of the arithmetics of natural numbers with that of p-adic integers could allow to avoid Gödel's problematics.

- Number theory looks from the point of view of TGD more fundamental than set theory and inspires the question whether the notion of algebraic number could emerge naturally from TGD. There are two ways to understand the emergence of algebraic numbers: the hierarchy of infinite primes in which ordinary primes are starting point and the arithmetics of Hilbert spaces with tensor product and direct sum replacing the usual arithmetic operations. Extensions of rationals give also rise to cognitive variants of n-D spaces.

- The notion of empty set looks artificial from the point of view of physicist and a possible cure is to take arithmetics as a model. Natural numbers would be analogous to nonempty sets and integers would correspond to pairs of sets (A,B), A⊂ B or B⊂ A with equivalence (A,B)== (A∪ C,B∪ C). Empty set would correspond to pairs (A,A). In quantum context the generalization of the notion of being member of set a∈ A suggests a generalization: being an element in set would generalize to being single particle state which in general is de-localized to the set. Subsets would correspond to many-particle states. The basic operation would be addition or removal of element represented in terms of oscillator operator. The order of elements of set does not matter: this would generalize to bosonic and fermionic many particle states and even braid statistics can be considered. In bosonic case one can have multiple points - kind of Bose-Einstein condensate.

- One can also start from finite-D Hilbert space and identify set as the collection of labels for the states. In infinite-D case there are two cases corresponding to separable and non-separable Hilbert spaces. The condition that the norm of the state is finite without infinite normalization constants forces selection of de-localized discrete basis in the case of a continuous set like reals. This inspires the question whether the axiom of choice should be given up. One possibility is that one can have only states localized to finite or at least discrete set of points which correspond points with coordinates in an extension of rationals.

**1. Geometric analog for Gödel's problematics**

Goedel's problematics involves statements which cannot be proved to be true or false or are simultaneously true and false. This problematics has also a purely geometric analog in terms of set theoretic representation of Boolean algebras when real topology is used but not when p-adic topology is used.

The natural idea is that Boolean algebra is realized in terms of open sets such that the negation of statement corresponds to the complement of the set. In p-adic topologies open sets are simultaneously also closed and there are no boundaries: this makes them and - more generally Stone spaces - ideal for realizing Boolean algebra set theoretically. In real topology the complement of open set is closed and therefore not open and one has a problem.

Could one circumvent the problem somehow?

- If one replaces open sets with their closures (the closure of open set includes also its boundary, which does not belong to the open set) and closed complements of open sets, the analog of Boolean algebra would consist of closed sets. Closure of an open set and the closure of its open complement - stament and its negation - share the common boundary. Statement and its negation would be simultaneously true at the boundary. This strange situation reminds of Russell's paradox but in geometric form.

- If one replaces the closed complements of open sets with their open interiors, one has only open sets. Now the sphere would represent statement about which one cannot say whether it is true or false. This would look like Gödelian sentence but represented geometrically.

This leads to an already familiar conclusion: p-adic topology is natural for the geometric correlates of cognition, in particular Boolean cognition. Real topology is natural for the geometric correlates of sensory experience.

- Gödelian problematics is encountered already for arithmetics of natural numbers although naturals have no boundary in the discrete topology. Discrete topology does not however allow well-ordering of natural numbers crucial for the definition of natural number. In the induced real topology one can order them and can speak of boundaries of subsets of naturals. The ordering of natural numbers by size reflects the ordering of reals: it is very difficult to think about discrete without implicitly bringing in the continuum.

For p-adic integers the induced topology is p-adic. Is Gödelian problematics is absent in p-adic Boolean logic in which set and its complement are both open and closed. If this view is correct, p-adic integers might replace naturals in the axiomatics of arithmetics. The new element would be that most p-adic integers are of infinite size in real sense. One has a natural division of them to cognitively representable ones finite also in real sense and non-representable ones infinite in real sense. Note however that rationals have periodic pinary expansion and can be represented as pairs of finite natural numbers.

**2. Number theory from quantum theory**

Could one define or at least represent the notion of number using the notions of quantum physics? A natural starting point is hierarchy of extensions of rationals defining hierarchy of adeles. Could one obtain rationals and their extensions from simplest possible quantum theory in which one just constructs many particle states by adding or removing particles using creation and annihilation operators?

**2.1 How to obtain rationals and their extensions?**

Rationals and their extensions are fundamental in TGD. Can one have quantal construction for them?

- One should construct rationals first. Suppose one starts from the notion of finite prime as something God-given. At the first step one constructs infinite primes as analogs for many-particle states in super-symmetric arithmetic quantum field theory. Ordinary primes label states of fermions and bosons. Infinite primes as the analogs of free many-particle states correspond to rationals in a natural manner.

- One obtains also analogs of bound states which are mappable to irreducible polynomials, whose roots define algebraic numbers. This would give hierarchy of algebraic extensions of rationals. At higher levels of the hierarchy one obtains also analogs of prime polynomials with number of variables larger than 1. One might say that algebraic geometry has quantal representation. This might be very relevant for the physical representability of basic mathematical structures.

**2.2 Arithmetics of Hilbert spaces**

The notions of prime and divisibility and even basic arithmetics emerge also from the tensor product and direct sum for Hilbert spaces. Hilbert spaces with prime dimension do not decompose to tensor products of lower-dimensional Hilbert spaces. One can even perform a formal generalization of the dimension of Hilbert space so that it becomes rational and even algebraic number.

For some years ago I indeed played with this thought but at that time I did not have in mind reduction of number theory to the arithemetics of Hilbert spaces. If this really makes sense, numbers could be replaced by Hilbert spaces with product and sum identified as tensor product and direct sum!

Finite-dimensional Hilbert space represent the analogs of natural numbers. The analogs of integers could be defined as pairs (m,n) of Hilbert spaces with spaces (m,n) and (m+r,n+r) identified (this space would have dimension m-n. This identification would hold true also at the level of states. Hilbert spaces with negative dimension would correspond to pairs with (m-n)<0: the canonical representives for m and -m would be (m,0) and (0,m). Rationals can be defined as pairs (m,n) of Hilbert spaces with pairs (m,n) and (km,kn) identified. These identifications would give rise to kind of gauge conditions and canonical representatives for m and 1/m are (m,1) and (1,m).

What about Hilbert spaces for which the dimension is algebraic number? Algebraic numbers allow a description in terms of partial fractions and Stern-Brocot (S-B) tree (see this and this) containing given rational number once. S-B tree allows to see information about algebraic numbers as constructible by using an algorithm with finite number of steps, which is allowed if one accepts abstraction as basic aspect of cognition. Algebraic number could be seen as a periodic partial fraction defining an infinite path in S-B tree. Each node along this path would correspond to a rational having Hilbert space analog. Hilbert space with algebraic dimension would correspond to this kind of path in the space of Hilbert spaces with rational dimension. Transcendentals allow identification as non-pediodic partial fraction and could correspond to non-periodic paths so that also they could have Hilbert spaces counterparts.

**2.3 How to obtain the analogs higher-D spaces?**

Algebraic extensions of rationals allow cognitive realization of spaces with arbitrary dimension identified as algebraic dimension of extension of rationals.

- One can obtain n-dimensional spaces (in algebraic sense) with integer valued coordinates from n-D extensions of rationals. Now the n-tuples defining numbers of extension and differing by permutations are not equivalent so that one obtains n-D space rather than n-D space divided by permutation group S
_{n}. This is enough at the level of cognitive representations and could explain why we are able to imagine spaces of arbitrary dimension although we cannot represent them cognitively.

- One obtains also Galois group and orbits of set A of points of extension under Galois group G as G(A). One obtains also discrete coset spaces G/H and alike. These do not have any direct analog in the set theory. The hierarchy of Galois groups would bring in discrete group theory automatically. The basic machinery of quantum theory emerges elegantly from number theoretic vision.

- In octonionic approach to quantum TGD one obtains also hierarchy of extensions of rationals since space-time surface correspond zero loci for RE or IM for octonionic polynomials obtained by algebraic continuation from real polynomials with coeffficients in extension of rationals (see this).

**3. Could quantum set theory make sense?**

In the following my view point is that of quantum physicist fascinated by number theory and willing to reduce set theory to what could be called called quantum set theory. It would follow from physics as generalised number theory (adelic physics) and have ordinary set theory as classical correlate.

- From the point of quantum physics set theory and the notion of number based on set theory look somewhat artificial constructs. Nonempty set is a natural concept but empty set and set having empty set as element used as basic building brick in the construction of natural numbers looks weird to me.

- From TGD point of view it would seem that number theory plus some basic pieces of quantum theory might be more fundamental than set theory. Could set theory emerge as a classical correlate for quantum number theory already considered and could quantal set theory make sense?

**3.1 Quantum set theory**

What quantum set theory could mean? Suppose that number theory-quantum theory connection really works. What about set theory? Or perhaps its quantum counterpart having ordinary set theory as a classical correlate?

- A purely quantal input to the notion of set would be replacement of points delocalized states in the set. A generic single particle quantum state as analog of element of set would not be localized to a single element of set. The condition that the state has finite norm implies in the case of continuous set like reals that one cannot have completely localized states. This would give quantal limitation to the axiom of choice. One can have any discrete basis of state functions in the set but one cannot pick up just one point since this state would have infinite norm.

The idea about allowing only say rationals is not needed since there is infinite number of different choices of basis. Finite measurement resolution is however unvoidable. An alternative option is restriction of the domains of wave functions to a discrete set of points. This set can be chosen in very many manners and points with coordinates in extension of rationals are very natural and would define cognitive representation.

- One can construct also the analogs of subsets as many-particle states. The basic operation would be addition/removal of a particle from quantum state represented by the action of creation/annihilation operator.

Bosonic states would be invariant under permutations of single particle states just like set is the equivalence class for a collection of elements (a

_{1},...,a_{n}) such that any two permutations are equivalent. Quantum set theory would however bring in something new: the possibility of fermionic statistics. Permutation would change the state by phase factor -1. One would have fermionic and bosonic sets. For bosonic sets one could have multiple elements ("Bose-Einstein condensation"): in the theory of surfaces this could allow multiple copies of the same surface. Even braid statistics is possible. The phase factor in permutation could be complex. Even non-commutative statistics can be considered.

Many particle states formed from particles, which are not identical are also possible and now the different particle types can be ordered. On obtains n-ples decomposing to ordered K-ple of n

_{i}-ples, which are consist of identical particles and are quantum sets. One could talk about K-sets as a generalization of set as analogs of classical sets with K-colored elements. Group theory would enter into the picture via permutation groups and braid groups would bring in braid statistics. Braids strands would have K colors.

**3.2 How to obtain classical set theory?**

How could one obtain classical set theory?

- Many-particle states represented algebraically are detected in lab as sets: this is quantum classical correspondence. This remains to me one of the really mysterious looking aspects in the interpretation of quantum field theory. For some reason it is usually not mentioned at all in popularizations. The reason is probably that popularization deals typically with wave mechanics but not quantum field theory unless it is about Higgs mechanism, which is the weakest part of quantum field theory!

- From the point of quantum theory empty set would correspond to vacuum. It is not observable as such. Could the situation change in the presence of second state representing the environment? Could the fundamental sets be always
*non-empty*and correspond to states with non-vanishing particle number. Natural numbers would correspond to eigenvalues of an observable telling the cardinality of set. Could representable sets be like natural numbers?

- Usually integers are identified as pairs of natural numbers (m,n) such that integer corresponds to m-n. Could the set theoretic analog of integer be a pair (A,B) of sets such that A is subset of B or vice versa? Note that this does not allow pairs with disjoint members. (A,A) would correspond to empty set. This would give rise to sets (A,B) and their "antisets" (B,A) as analogs of positive and negative integers.

One can argue that antisets are not physically realizable. Sets and antisets would have as analogs two quantizations in which the roles of oscillator operators and their hermitian conjugates are changed. The operators annihilating the ground state are called annilation operators. Only either of these realization is possible but not both simultaneously.

In ZEO one can ask whether these two options correspond to positive and negative energy parts of zero energy states or to the states with state function reduction at either boundary of CD identified as correlates for conscious entities with opposite arrows of geometric time (generalized Zeno effect).

- The cardinality of set, the number of elements in the set, could correspond to eigenvalue of observable measuring particle number. Many-particle states consisting of bosons or fermions would be analogs for sets since the ordering does not matter. Also braid statistics would be possible.

What about cardinality as a p-adic integer? In p-adic context one can assign to integer m, integer -m as m× (p-1)× (1+p+p

^{2}+...). This is infinite as real integer but finite as p-adic integer. Could one say that the antiset of m-element as analog of negative integer has cardinality -m= m(p-1)(1+p+p^{2}+..). This number does not have cognitive representation since it is not finite as real number but is cognizable.

One could argue that negative numbers are cognizable but not cognitively representable as cardinality of set? This representation must be distinguished from cognitive representations as a point of imbedding space with coordinates in extension of rationals. Could one say that antisets and empty set as its own antiset can be cognized but cannot be cognitively represented?

Here one has two views about infinite set. One has both separable (infinite-D in discrete sense: particle in box with discrete momentum spectrum) and non-separable (infinite-D in real sense: free particle with continuous momentum spectrum) Hilbert spaces. In the latter case the completely localized single particle states would be represented by delta functions divided by infinite normalization factors. They are routinely used in Dirac's bra-ket formalism but problems emerge in quantum field theory.

A possible solution is that one weakens the axiom of choice and accepts that only discrete points set (possibly finite) are cognitively representable and one has wave functions localized to discrete set of points. A stronger assumption is that these points have coordinates in extension of rationals so that one obtains number theoretical universality and adeles. This is TGD view and conforms also with the identification of hyper-finite factors of type II_{1} as basic algebraic objects in TGD based quantum theory as opposed to wave mechanics (type I) and quantum field theory (type III). They are infinite-D but allow excellent approximation as finite-D objects.

This picture could relate to the notion of non-commutative geometry, where set emerges as spectrum of algebra: the points of spectrum label the ideals of the integer elements of algebra.

See the article Some layman considerations related to the fundamentals of mathematics.

For a summary of earlier postings see Latest progress in TGD.

## 13 Comments:

You have approached our question largely in language of Topology, which I'm not familiar with. But here's a very interesting lecture about foundational issues in that language, and from what I was able to gather, convays much of the same meaning as <> is attempting to open and reveal:

https://www.youtube.com/watch?v=sDGnE8eja5o

The final question in the lecture, about practical setup for doing approximate geometry is where "set" or "self" consisting of relational operators <> could show its real power. On this foundational level "approximate" should be of course be understood to mean same as 'prediscreet'/'prequantified', as 'approximate' is relative, codependent opposition to the notion of 'exactly quantified'. Most basic and universal non-sine-qua generative structure underlying all geometry and number theory is maintained, but the problem vanishes as it is not created in the first place by starting from quantification, different versions of which can be derived and/or constructed as practical and effective subtheories.

Also, interpreting <> as the most general and fundamental generator of 'causal diamonds' seems very natural.

Let us now remember the real origin of birth of post-Cartesian science, which is rigorous Cartesian skepticism, concluding that the empirical reality that cannot be doubted is the experiental content of 'self' at a given moment, meaning in this formulation inside <>. The true evolutionary meaning of a Cartesian moment of self-realization is however inherently marked at the dynamic open boundaries: What Else? What More, what Less?

Topology is important part of TGD. But only one part. One could also see TGD in terms of differential geometry of sub-manifolds in M^4xCP_2 picture. Or as algebraic geometry in M^8 picture.

Postulating exact quantification is more or less assuming Axioms of Choice. This is definitely non-realistic. The challenge is to construct elegant theory of finite measurement resolution. This could be one starting point. Hyper-finite factors emerge at Hilbert space level and cognitive representation as points in extension of rationals common to various number fields emerge at the level of geometry. Note that also Hilbert space has extension of rationals as coefficient field.

In p-adic context <> is problematic for reasons which I have mentioned many times. Furthermore, the very feature of finite measurement resolution is that ordering is lost below measurement scale so that real number based approach carries un-necessary information and bits without meaning are always bad.

Empirical reality is defined in terms of conscious experience. We experience only change and from this manage to deduce the idea about what exists by using logic and something transcendental which cognitive representations are not able to represent. I believe that after century has passes the idea about continually re-created reality is taught in elementary schools;-). It provides the only manner to avoid paradoxes.

Physicist may think of primes as primitives, but for foundational pure mathematician prime needs to be defined by more primitive elements. With new foundational theory we are at liberty to think differently and try to reformulate alternative approach also to adelics and p-adics, without the extra weight and problems carried by the Cantorian set theory. If we want to think differently and hopefully more generally, the notion of closed metric infinity is excluded at this stage. We approch infinities with open generators. Therefore the standard definition of Hilbert space is something we avoid.

Suppose we have already quantified theory of integers as a sub of basic number-identity-antinumber scheme, we can then derive the "adelic" spine as Euler's identity of exponential multiplication:

<<< << < <1> <1/n^1>> <1/n^2>>> <1/n^3>>>>

AFAIK, we should be able to compress the information in this scheme by replacing n with p.

The form <> has it's own topology, that of pre-quantified affine geometry of parallel lines, on which Cartesian grids are based. So as generator <> is far more stronger and general than the round objects of the topology lecture linked above. Notion of area becomes naturally visible and definable already at this primitive affine level, we don't have to wait for the postulation of metric spaces to define it.

To my comprehension, the key notion of p-adics is that we don't start them by piling up and combining atomistic quantities, but by partioning unity, going from whole to parts. In this sense egyptian fractions in the form 1/n have 1-adic quality.

But what we are looking for is some geometric, prequantified intuition for the whole that can be partitioned p-adically. Natural candidate could be equilateral triangle (also implied in the form <>), for the first "genuine" prime, and then generalizing from that to regular n-gons.

Of course, these is just amateur tentative suggestion for a way to proceed from <> to a theory of p-adics, and real mathematician with better mind can take it from here, or leave it.

One can imagine many representations of prime. As regular polygons with prime number of edges or as finite fields Z_p labelled by primes. It is probably an easy exercise to show that any integer can be expressed as products of powers of primes so that successor axiom might allow elimination and one cold replaced the notion of Cantorian infinity with a number theoretically universal notion of infity as described by the hierarchy of infinite prime/integers/rationals. Infinite numbers have precisely defined number theoretical anatomy and their ratios and their real and p-adic norms are also well-defined. Physically this is much more interesting than Cantorian hierarchy which looks to me rather dull.

What is needed is precise definition of < and >.

My basic intuition is that it is best to keep relational OPERATORS < and > as the name says, as verbs that can do more and less. From the verb stage not just one but, as needed, more noun-like precise definitions can be given to study their structures and conclusions and interactions. I understand mathematics as general study of relations, and precise definitions are essential to be able to talk about relations without ambiguities. And I agree that precise definitions are much better way to proceed in the study of relations than messy axioms with their messy logicism.

There have been already many suggestions for possible definitions. As the motive for 'quantum set' has been to find simpler and better mathematical theory to describe QM than the current mess, obvious possibility to proceed towards finding useful definition is to try to define 'operator' in such a clear and precise manner that we can simplyfy and generalize the notion of 'operator' in QM.

Here's what Wiki says: "In physics, an operator is a function over a space of physical states to another space of physical states. The simplest example of the utility of operators is the study of symmetry."

Starting with just relational operators we do not need to presuppose space (or time) as anything predefined, but as the shape '<' suggest's in itself, we can take it as pair of "vectors" (written in quotes as intuitive notion and leaving more precise definition of vector for later), naming one "vector" as 'space' and other 'time'. The basic idea is that in this manner we can do spacetime <> as we go, and give spacetime richer form as we develop the theory further with more definitions, hoping that our foundational development connects also with empirical reality of existing standard theories of physics in a meaningful and beautiful way.

If this sounds reasonable approach to you, how could we proceed to define other operators needed for fuller description? Can this approach help to cut some corners and find new generalizations for the required tool pack of operators?

Another thought: There's more to the universe than we can measure.

In the basic scheme and hierarchy of geometries that I have learned, both prequantified and affine and projective geometry are primary to quantified metric geometries. If we accept the humble view that we observe universe from participant point of view, not externally from God's All-Knowing view, we can perhaps think of the the issue of locality and non-locality in a somewhat novel way. Instead of assuming that our locally cognitive quantified mathematics extends similarly to every region of the whole Universe, we could take quantified metrics as just local phenomena - local defined now in new more general manner - and prequantified affine and projective geometries as what is cognizable to us of the larger and inclusive non-local whole in some holographic manner. Arithmetic computation could in this sense be purely local phenomena, and algebraic relations the rich border zone between our local point of view and prequantified non-local whole.

The definition '<> contains area (or physically: spacetime) between something more, something less than itself' rings true in this sense. Interpretation of holographic hierarchy of selves, subselves as well as their peer rings follows naturally.

Relations are between objects and I do not believe that it makes much sense to have only relations between object which do not exist. This is like Copenhagen interpretation in which Psi is information about something which does not exist.

I see objects as rather natural things. Objects are quantum states and consciousness emerges via quantum jumps between them. State function reduction is analogous to relation but one cannot do without quantum states.

If one accepts the notion of Hilbert space as basics of quantum theory , also the notion of operator is well-defined as linear map between Hilbert spaces. The theory of Hilbert space operators is very simple and full of elegance. This is why I feel Bohmian approach as extremely awkward. If one gives up the notion of Hilbert space, one cannot talk about operators as anything well-defined.

I see mathematics as the practical art of identifying reasonable unprovable truthts called axioms and deducing more truths from these by rules of logic. If one assumes nothing one cannot deduce nothing. If there is no space or time, there is not much to be done except developing handweaving arguments about how they "emerge". This is the latest fad in theoretical physics and predictably has produced nothing interesting. One can construct ad hoc rules for how to form strings of < and > and put computer to consruct and print them but I do not understand what wisdom this would yield when there is no interpretation for thee output because the relations are between objects which do not exist.

The question is about the notion of number fields, to me they look part of fundamental mathematics. One can argue about details of axiomatics but this is about details. Axiomatics as such is necessary in order to keep books about basic assumptions, nothing bad in that when one recalls that the number of basic non-provable truths is infinite.

There is more than we can measure in the Universe. Here I can agree. I have been talking about cognitive representations as intersection of real and various p-adic worlds. Cognitive representations are only this intersection and consisting of common points of these worlds. Without the rest we would not have sensory world and cognition. At space-time level cognitive representation consists of points of space-time with imbedding space coordinates in extension of rationals defining the adele. At WCW level from surface with parameters in this extension. This view leads immediately to the identification of evolution as algebraic evolution towards more complex extensions.

So we come back to the most basic philosophical problem of metaphysics: assigning independent and inherent existence to objects. That belief sums up to the claim of having found the proverbial Archimedean point: https://en.wikipedia.org/wiki/Archimedean_point

For a general linguist who has read his de Saussure, the meaning is clear: Archimedes is pointing that he is not in possession of the Archimedean point, it is nowhere to be found out. Neither rigorous empirical observation nor consistent rational ('ratio' is Latin for relation) thinking have found any Archimedean point, object with inherent and independent existence. On the other hand it is easy to show that we can relate relations without postulating any subject-object relation: <>, simple Finnish asubjective sentences. "Suhteudutaan.", "Enenee." "Vähenee." etc.

The meaning of a sign (word, object, etc.) arises from it's relation to the whole network of relations. Looking at a dictionary, definitions of words consist of other words, the whole networkd of signs. Indra's Net. David Bohm, who AFAIK coined the holograph metaphor for this general principle, had deep comprehension of interdependent relativity of all phenomena. TGD also mentions 'Strong Holographic Principle', and occationally pays lip service to the codependent causality that Gautama taught: "If this arises, that arises, if this ceases, that ceases." But in the light of what you now say, that seems to be just some superficial decoration devoid of genuine meaning and comprehension. For a self-important believer nothing is as important as believing that his beliefs are important. More important than empiricism, rationalism, relating as fellow sentient beings. More important than wisdom.

Difference between 'axiom' and 'definition' is subtle, but all important. Axioms are about giving truth values to believe in, definitions don't presuppose truth values but are given as honest attempts to communicate and discuss and think as clearly as we can.

You are familiar with the story about Joukahainen and Väinämöinen. Subjective Joukahainen "luulee tietävänsä" (reckons knowing), boasts arrogantly knowing the axioms of firm ontological objects, boasts knowing the Archimedean point, the One Ring To Rule Them All. Asubjective, holographic and holonomic Väinämöinen "tietää luulevansa" (knows reckoning), and can therefore teach Joukahainen a good lesson, transform what are only reckonings, sing away the (not-so!) firm ground beneath Joukahainen's feet, so that Joukahainen sinks into the empirically undeniable bog-of-no-firm-point.

The myth of Joukahainen and Väinämöinen continues to even more interesting topic, the meaning of Unique (Aino), and how unique is something that one can only be, not possess. But instead of retelling that story, the basic meaning is expressed more simply and eloquently in the Lakota prayer:

Aho Mitakuye Oyasin - To All Our Relations.

https://www.academia.edu/34793448/The_Fork_in_the_Mathematical_Road_Introduction_Part_1_of_3_

Thanks for a lesson in philosophy. I must admit that colleagues who hate philosophy have some justification for their attitude. If philosophy is something like this, physics can quite well do without it. My view about philosophical thinking, which I very greatly appreciate, is very different.

Of course, your obvious goal is to a to deprive value from TGD in your eyes, to see yourself as the Old and Wise Väinämöinen and me as the young and stupid Joukahainen. You ego have right to do this self deception in its attempt to reduce its suffering: it is your narrative. Not very great one as postmodernist would perhaps say.

I am a theoretical physicst. In this job one cannot waste time to product this kind of world salad as you did. I am busily developing the theory and applying it.

a) In physics must make assumption about objects. Just starting to talk about < and > definining relations between non-existing objects and without any definition for these arrows is from this point of view romantic sounding pseudophilosophical non-sense. I promise to update my view if you can apply this approach to some physical problem, say dark matter problem. Saying that neither dark matter nor matter do not exist since nothing exists is not a solution to the problem.

A slight inaccuracy in your text: "vähenee" does not describe a relation which always has two members A and B. You must give some other name to your > and <.

b) In TGD objects are purely mathematical ones: zero energy states and there is no further physical reality behind them. Conscious experience is second form of existence and corresponds to quantum jumps between them. I strongly suggest that you carefully re-read this sentence for few times and consider that it eliminates physical existence as something behind the mathematical existence from the ontology.

c) The basic idea of TGD inspired consciousness is to give up of the notion of observer as outsider to the physical world and making it part of it with an extension of ontology to include also state function reductions as the basic building brick of consciousness. Against this fact, the accusation about postulating Archimedean vantage point only shows that you have not understood or even tried to understand what I am saying.

This of course also brings relativity. Conscious experience and therefore theories giving very limited information about physical existence (zero energy states recreated all the time, not single reality) are subjective and are bound to be more or less wrong. Some of them are however more wrong than others and some not even wrong.

d) I gave up my attempts to understand the burst of accusations for which "Gautama" serves as a convenient label. "Interdependent relativity" appears in it. The already mentioned relativity is basic aspect in recent day theoretical physics but has precise content.

The burst contains an imprecise statement: To my opinion "In light what you say..." should be replaced with "In light of what I understand about what you say - recalling the limitations posed by lacking basic education in mathematics and physics - ...".

e) Concerning axioms and definition: you get nowhere if you have only definitions. You cannot make a revolution in mathematics without having some understanding of mathematics. If you want to revolutionize mathematics, why not to start a systematic serious study of mathematics and physics? It might take 5-10 years to learn basics of mathematics and physics. We have known for more than decade. If you had started the process already decade ago, our discussions could be much more rewarding.

Matti.. I am so glad you are now contemplating the role...either direct or historical of P-Adics,,, As you know I have begun to speculate about P-Adic and the counterproductive straightjacketing of time with the three spatial dimensions....... in a Euclidean snare... ..from the more philosophical level of the poetry of our intuitions......

This essay is soon expanding considerably...but would appreciate any comments you have on it. Thanks

https://www.academia.edu/34868376/Freeing_Time_from_Euclids_Grasp_Numbers_and_Geometry_

https://en.wikipedia.org/wiki/Memristor

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