Dear Arved Huebler,

thanks for your comment. I think what HoTT does, is to facilitate a formal system which in its inner core states that without consistency there is no existence possible. If one assumes HoTT to be consistent itself, this leads to the impression that HoTT must capture and represent the fundamental truth. But not all consistent schemes do necessarily meet reality. Maybe HoTT does answer this question in the positive and says that all mathematically consistent formal systems are to be generated at some point (in time) and therefore indeed meet reality.

I think HoTT does state a triviality, namely that inconsistencies do not pave the path from the abstract to the concrete (especially physical reality). This path should symbolize the well known deductive principle in logics. Now, HoTT makes a huge leap by saying that if something suffices consistency, it must exist (somewhere, somehow). If one does understand HoTT as merely stating that consistency is necessary for existence, but not sufficient, i would wonder what the needed sufficient additional properties are to make something physically existent.

By assuming an entity of 'non-existence', and by stating that only consistent structures can come into existence, this 'non-existence' must be considered as existent - because within HoTT, it has the feature of consistency. Otherwise it could never produce the whole chain of events that led to our universe. Stated differently: the entity of 'non-existence' has a single property, namely the potential to produce something. This is a somewhat trivial logical conclusion based on our factual existence. Therefore one must understand your entity of 'non-existence' as existent. How can one and the same thing be existent and non-existent at the same instant? I really think that we cannot logically consistent assume such a non-existent entity being the first unmoved mover. And because this is logically inconsistent to me, i think, it must fail, although i nonetheless do not exclude that mathematics was generated by some other causes than itself.

Interesting essay. It has some overlap with the FQXi essay I wrote in 2015. I discuss HOTT there. This does seem to be a discrete structure that emerges from homotopy theory that is a digital-like system of types. My current essay does discuss some math foundation issues as well.

It is interesting to ponder the nature between the continuum with its infinitesimals and points with the finite and discrete. The continuum has no physical content as I see it, but is a convenient model system for a lot of physics. The physical dot or pixel (voxel) in quantum gravity is the Planck length, and this is the smallest length or region where a qubit of information can be localized. Anything smaller than this is the continuum that has a somewhat different physical meaning, and in the case of infinitesimals the physical meaning reduces to zero.

Cheers LC

    Arved C. Huebler,

    Quoting you, "The mathematical empty set is like a substrate, a carrier or a homogeneous background. The empty set is the precondition for the existence,"

    The empty set (substrate) thus seems to be not of mathematical origin. Your analogy of 'pixels' creating dots on 'paper' using 'ink' is interesting. Here, there are three players. Which of these are of mathematical origin? In my opinion, the pixels alone are mathematical. The 'paper and ink' together create a 'black and white' substrate; if paper is white, ink is black, if paper is black ink is white. Only then can the pixels create dots. So logically 'ink' is also not of mathematical origin. So I argue that the substrate 'paper and ink' is of physical origin and 'pixels' is of mathematical origin. So what we see as 'dots' and 'complex patterns of dots' are just mathematical possibilities of the physical world. Physics provides the substrate, mathematics decides the structures.

    Jose P Koshy

      Dear Arved,

      Your approach is interesting and I agree with some of your points. You talk about information but how do you give sense to that information ? Someone or something should to be observing it ? Information alone is just potential information, I believe that some sort of consciousness needs to be brought into the equation in order to transform that potential information into "real" information and eventually create a reality.

      If you feel like it, please take a look at my essay and you will see more precisely what I mean. (A Universe of information and consciousness).

      All the best,

      Patrick

        Dear Stefan Weckbach,

        Thanks for your comprehensive reply, which helped me a lot.

        To be frank, I am not in a position to judge HoTT and understand it in all aspects. Like a lot of other formal constructions, as you mentioned it shall consist of trivial elements, too. What I learned and where I was inspired is the idea of designing mathematics as a process instead of a steady structure, as the traditional understanding suggests. The motivation for their approach was, as I understood the story of HoTT, to receive the ability to verify mathematical statements automatically.

        It is only a suggestion, a hypothesis. And if mathematics is a process, which is enfolding a mathematical universe, we can ask about possible effects. My result for this question is not a homogeneous mathematical universe, but a singularity, which has to become happen.

        The other question is about the definition of mathematics itself, whether it is formulated as HoTT, as a set theory or what ever. I think, my argumentation in the essay is to short, and my explanation is not clear enough:

        Most of the scientists follow Platon with his differentiation between idea and form, which means today: Only physics is true reality, mathematics is something else. If you can't measure, it is not reality. As I understood, you argue in this way, too.

        My hypothesis: Lets assume, there is no difference between idea and form. In this case, what mathematics might represent? It should be the „existence" itself as a "pre-physical" entity.

        You are right, my argument of logical induction from an initial point of non-existence towards an unfolding complex structure is simple, as the HoTT is simple in its basic foundation. But is it wrong? I have the vague hope, that it might be possible to generate a mathematical estimation for the parameters of the assumed singularity. And perhaps, at the end this might offer a link to physical entities like physical space and time.

        And finally, you have criticized my argumentation regarding non-existence. Perhaps it was wrong to confuse the reader with this more philosophical stuff within this few eight pages. The idea was to define the initial starting point of mathematics. It can be the existence itself, which is unfolded to the infinite complex structure of mathematics. But if you assume non-existence as first entity ever, in the same moment existence is logically included as a second entity. For the main hypotheses, this question seems not crucial.

        Dear Arved C. Huebler

        I invite you and every physicist to read my work "TIME ORIGIN,DEFINITION AND EMPIRICAL MEANING FOR PHYSICISTS, Héctor Daniel Gianni ,I'm not a physicist.

        How people interested in "Time" could feel about related things to the subject.

        1) Intellectuals interested in Time issues usually have a nice and creative wander for the unknown.

        2) They usually enjoy this wander of their searches around it.

        3) For millenniums this wander has been shared by a lot of creative people around the world.

        4) What if suddenly, something considered quasi impossible to be found or discovered such as "Time" definition and experimental meaning confronts them?

        5) Their reaction would be like, something unbelievable,... a kind of disappointment, probably interpreted as a loss of wander.....

        6) ....worst than that, if we say that what was found or discovered wasn't a viable theory, but a proved fact.

        7) Then it would become offensive to be part of the millenary problem solution, instead of being a reason for happiness and satisfaction.

        8) The reader approach to the news would be paradoxically adverse.

        9) Instead, I think it should be a nice welcome to discovery, to be received with opened arms and considered to be read with full attention.

        11)Time "existence" is exclusive as a "measuring system", its physical existence can't be proved by science, as the "time system" is. Experimentally "time" is "movement", we can prove that, showing that with clocks we measure "constant and uniform" movement and not "the so called Time".

        12)The original "time manuscript" has 23 pages, my manuscript in this contest has only 9 pages.

        I share this brief with people interested in "time" and with physicists who have been in sore need of this issue for the last 50 or 60 years.

        Héctor

        Dear Conrad,

        thanks for your feedback and impetuses.

        If I understand right, you feel mathematics is to simple to cover all the phenomena of physics, e.g. continuous fields. But I thing, it is possible to show mathematically a transition from a countable set of infinite entities to a continuous entity.

        Perhaps my text was not sufficiently intelligible. The term pixel was defined as a basic element, which can be identified at a certain scale level. This pixel may consist of very complex substructures, based again on basic elements at a lower scale, and so on. So, an printed dot of ink is much more complex than one quantum dot, because it consists of an uncountable number of quantum dots in a very complex structure.

        As you state very right, a ordinary logical (=: mathematical in my essay) structure itself has no effect in the physical word, it needs an entity with at least one physical parameter to effect something in the physical world.

        But my hypothesis is: At a fare end of highly complex mathematical structures, an emergent creation of a first physical entity, e.g. a Minkowski cell, might happen. And the I look for a possible way, how this can work.

        Regards

        Arved

        Dear Sridattadev Kancharla,

        thanks for your feedback, but I am to restricted to follow you argumentation. Fpr me it is difficult to find the link between your approach and my essay.

        Good luck for you work

        Regards

        Arved Hübler

        Dear Eckard,

        thanks for you statement. But what a pity, the vice versa is my approach.

        Lets try to explain my understanding by and improve the weak explanation at my essay:

        Assume elements, and assume these elements will assemble a pattern, a structure. You may be able to discover logical rules and regularities in this structures. If the elements have physical features, you will discover physical laws. If we think the elements without any physical properties as an abstract model, we will discover pure mathematics, logical patterns itself. That is, what you agree about.

        Now the vice versa:

        My 1st hypothesis: This pure mathematics may exist without any physics. They exist as logical pattern of the existence itself (I call it mathematical point). You do not agree, as you stated.

        But it is only a hypothesis to see what happens, if we assume this. Is there a logically reason, why this assumption might be nonsense?

        My 2nd hypothesis: At a moment when this pure mathematics becomes highly complex, an emergent step will create a first physical entity (e.g. a Minkowsky cell or something else).

        Again: Is there a logically reason, why this assumption might be nonsense?

        Rest of my essay is asking for A possible explanation, how and why such an emergent process works. I supposed a singularity.

        My future task: To proof this hypothesis (and perhaps adjust it).

        But if there are arguments showing that these hypotheses are impossible, I could save time.

        Thanks for your time and regards

        Arved

        Dear Héctor Daniel Gianni,

        thank your for your long list of statements. Unfortunately I am not able to find a connection to my essay, so I am not in the position to comment it.

        Thanks again and good luck.

        Regards

        Arved

        Dear Lawrence,

        thanks for your statement. I have read your earlier paper "Mathematical Physics as Topological Numbers, Types and Quanta" and I found a lot of interesting ideas. But for this essay of mine, I decided not to review other papers because of the eight pages restriction. I am very sorry.

        For sure, the question of the physical content of an continuum is interesting, as you discussed in your paper. But I believe, it is not such a big secret. Perhaps it is related to the emergent step from one scale to another.

        Also interesting is the physical equivalent of the smallest physical dot, as you mentioned. I did not proceed as far as quantum information, because that is a very tricky discussion, which I want to avoid. What is a technical necessity at a qubit, and what is the real physically effective information content?

        So, I restrict myself and talk only about bits in general, which also includes qubits. And in my essay I used the term "pixel", because that is a single parameter entity. Your suggestion to talk about voxels refers to three dimensions, but perhaps one dimension is the initial state.

        Thanks again,

        Regards

        Arved

        Dear Jose,

        thank you for your contribution. You highlight a very interesting question, which was the initial motivation of mine to enter this field between physics an information and mathematics.

        But I do not agree with your way to distinguish mathematics from physics. All three 鈥瀙layer", as you called it, have an equivalent entity either in the mathematics and the physics:

        name - physical entity, e.g.print -聽mathematical entity

        existence - printed dot on paper -聽 mathematical point/element

        potential existence - unprinted paper -聽 empty set

        impossible existence -聽missing sheet -聽 not defined

        Acording to my hypothesis, mathematics describes only the existence itself and its logical relations (mathematical rules). But in physics we have the existence of at least one physical parameter, e.g. voltage or spin, or here in the case of printing the visual contrast, and its logical relations (physical laws).

        Regards

        Arved

        Dear Patrick,

        thanks for your comments.

        Not I have to give sense to information, but something has to be able to sense a change, if this change should be information.

        I known, there are a lot of very complex definitions of information available, e.g. the term of "potential information" you have mentioned. But I want to start with the simplest definition of information, which is possible. And that is the "message of a change" for the case, that this change has an effect.

        But my essay was only able to address the way from the basic mathematical existence to the emergence point of physics. The way onwards to more complex forms of information like consciousness is not covered. A task for the future.

        Thanks and regards

        Arved

        Dear Shaikh Raisuddin,

        I see no relation to my essay, but from my point of view an answer to your question is: At least the system needs the capability to detect the direction of the goal.

        Regards

        Arved

        Dear Arved Huebler,

        thank you also for your reply.

        If mathematics has evolved from some empty set, at first glance it seems as if the rest of mathematics follows necessarily. The latter should be indeed true, because otherwise one couldn't trace it back to its origin - namely to the empty set. But what does this demand of necessity imply? For me, it implies that the empty set somehow contains already the whole of mathematics. But this would mean that mathematics didn't evolve, but was already hidden in the empty set. This would further mean that the empty set is not really an empty set, but could be identified with the whole of mathematics. Surely, mathematics could also have been arisen out of the blue and once it exists with its main properties (namely being a network of necessary relationships between its consituents), these relationships must be considered as necessary.

        But let's prove this a bit more in detail. Think about the number Pi. In the case of mathematics having evolved out of an empty set, its value 3.14159... is not a necessity, it could well be 4.14159... or any other string of digits. Because, as outlined above, necessity implies that the whole of mathematics exists already hidden in the empty set. Surely, if the value of Pi would be different from our known value, all the other mathematical relations also had to be other than they are to guarantee the consistency of maths and/or the consistency with our physical theories. The question here is how the value of 3.14159... comes about, if there is no maths around in an empty set. If this value is logically necessary, on what basis other than on the relation between the circumference and the diameter of a circle does it 'emerge'? If true, how did this circle have been emerged from an empty set? But if the existence of circles and alike are not logically necessary, what has determined their shape and values? It cannot be mathematics itself, because from the point of view of an empty set, there is no maths around anyhwere other than potential being (of some kind). So it seems to me that in both cases, the necessity of such mathematical values or their sheer 'randomness', their has to be some kind of existence beyond mathematics which has determined such values, either intentionally and / or by necessity, or randomly. The same is valid for the existence of circles and all kinds of geometrical shapes. Randomness cannot have achieved this, because without mathematics there is no definition of randomness (in an empty set). And on what ontological properties should this 'randomness' be based, i am forced to ask (if it really does exist, what i doubt).

        This leads me to the conclusion that if mathematics has somehow emerged, this could be only possible due to some entity that has more intelligence than mathematics itself. Mathematics, due to its inherent properties, cannot explain how it came about, because it cannot differentiate between a necessity and a possibility. If its existence is logically necessary, how can this be justified other than by a circular argument with the known existence of mathematics? And if it was just possible that it exists in the form we know it, what mechanism / or entity has decided which possibility to choose?

        You wrote

        "And finally, you have criticized my argumentation regarding non-existence. Perhaps it was wrong to confuse the reader with this more philosophical stuff within this few eight pages. The idea was to define the initial starting point of mathematics. It can be the existence itself, which is unfolded to the infinite complex structure of mathematics. But if you assume non-existence as first entity ever, in the same moment existence is logically included as a second entity. For the main hypotheses, this question seems not crucial."

        I agree, as far as the natural numbers are concerned. But for all other features of mathematics, i would pose the questions i wrote above. Anyways, thank you very much for your detailed reply and good luck in the contest!

        Best wishes,

        Stefan Weckbach

        Dear Professor Arved C. Huebler,

        Please excuse me for I have no intention of disparaging in any way any part of your essay.

        I merely wish to point out that "Everything should be made as simple as possible, but not simpler." Albert Einstein (1879 - 1955) Physicist & Nobel Laureate.

        Only nature could produce a reality so simple, a single cell amoeba could deal with it.

        The real Universe must consist only of one unified visible infinite physical surface occurring in one infinite dimension, that am always illuminated by infinite non-surface light.

        A more detailed explanation of natural reality can be found in my essay, SCORE ONE FOR SIMPLICITY. I do hope that you will read my essay and perhaps comment on its merit.

        Joe Fisher, Realist

        Nice essay Huebler,

        Your ideas and thinking are excellent for eg...

        At this macroscopic systems complexity and singularities play an important role. The idea is obvious to identify the singularity M as a candidate for the emergence of the physical universe.

        But singularities are mathematical problems, do you feel it is necessary that they exist in the real universe also?............

        ..................... At this point I want you to ask you to please have a look at my essay, where ...............reproduction of Galaxies in the Universe is described. Dynamic Universe Model is another mathematical model for Universe. Its mathematics show that the movement of masses will be having a purpose or goal, Different Galaxies will be born and die (quench) etc...just have a look at my essay... "Distances, Locations, Ages and Reproduction of Galaxies in our Dynamic Universe" where UGF (Universal Gravitational force) acting on each and every mass, will create a direction and purpose of movement.....

        I think intension is inherited from Universe itself to all Biological systems

        For your information Dynamic Universe model is totally based on experimental results. Here in Dynamic Universe Model Space is Space and time is time in cosmology level or in any level. In the classical general relativity, space and time are convertible in to each other.

        Many papers and books on Dynamic Universe Model were published by the author on unsolved problems of present day Physics, for example 'Absolute Rest frame of reference is not necessary' (1994) , 'Multiple bending of light ray can create many images for one Galaxy: in our dynamic universe', About "SITA" simulations, 'Missing mass in Galaxy is NOT required', "New mathematics tensors without Differential and Integral equations", "Information, Reality and Relics of Cosmic Microwave Background", "Dynamic Universe Model explains the Discrepancies of Very-Long-Baseline Interferometry Observations.", in 2015 'Explaining Formation of Astronomical Jets Using Dynamic Universe Model, 'Explaining Pioneer anomaly', 'Explaining Near luminal velocities in Astronomical jets', 'Observation of super luminal neutrinos', 'Process of quenching in Galaxies due to formation of hole at the center of Galaxy, as its central densemass dries up', "Dynamic Universe Model Predicts the Trajectory of New Horizons Satellite Going to Pluto" etc., are some more papers from the Dynamic Universe model. Four Books also were published. Book1 shows Dynamic Universe Model is singularity free and body to collision free, Book 2, and Book 3 are explanation of equations of Dynamic Universe model. Book 4 deals about prediction and finding of Blue shifted Galaxies in the universe.

        With axioms like... No Isotropy; No Homogeneity; No Space-time continuum; Non-uniform density of matter(Universe is lumpy); No singularities; No collisions between bodies; No Blackholes; No warm holes; No Bigbang; No repulsion between distant Galaxies; Non-empty Universe; No imaginary or negative time axis; No imaginary X, Y, Z axes; No differential and Integral Equations mathematically; No General Relativity and Model does not reduce to General Relativity on any condition; No Creation of matter like Bigbang or steady-state models; No many mini Bigbangs; No Missing Mass; No Dark matter; No Dark energy; No Bigbang generated CMB detected; No Multi-verses etc.

        Many predictions of Dynamic Universe Model came true, like Blue shifted Galaxies and no dark matter. Dynamic Universe Model gave many results otherwise difficult to explain

        Have a look at my essay on Dynamic Universe Model and its blog also where all my books and papers are available for free downloading...

        http://vaksdynamicuniversemodel.blogspot.in/

        Best wishes to your essay.

        For your blessings please................

        =snp. gupta

        Greetings Arved,

        I was attracted to this essay, as I am also a fan of the HoTT program, and I am glad I took the time to read it. I like the idea you are presenting and the first few pages of the paper very much. The way the paper wraps up fails to give the reader a clear idea of how this thesis fits into the assigned topic, and honestly it gets a little scattered, but it is an interesting exploration in its own right.

        Your comments on page 3 "The impossible existence (_) leads logically to its opposite, the possible or potential existence (т--Л). From this, real existence is derived (т--П), and then two existences are logically true (т--Пт--П)." are a precise re-statement of the ancient Chinese philosophical doctrine of Wu Ji and Tai Ji as taught by Taoist philosophers. In my essay, I argue that some of the ancient philosophical texts might encode secrets put there by mathematicians.

        But what you describe is broadly supportive of the main thesis in my own essay, Putting the Elephants to Work where I assert that evolutive trends observed in nature arise from properties of higher Mathematics. I would invite your comments. After reading your essay; I have the impression that HoTT offers rigorous proofs for some of the statements I made as logical arguments in my paper. But my work is also highly speculative, so it might be nice to know where my ideas are well supported.

        All the Best,

        Jonathan

          As it turns out..

          I got to spend a lot of time hanging out with printing presses as well.

          Warm Regards,

          Jonathan