Essay Abstract

Kant's transcendental philosophy is used to answer the questions about the nature of mathematics, how mathematics relates to the physical world, why we are self-aware and perceive ourselves in the world described by mathematics. Kant's architectonic system of reason is used to derive the invariant framework of the mind within which our thoughts originate - the original synthetic unity of apperception. It is the logical framework underlying all our possible knowledge - the framework of the cognitive faculty of understanding. This framework lies at the foundation of our thinking, logic, mathematics, natural language and organization of sense-data (experience). Phenomenal world in space and time is an output of this framework after the synthesis of the productive imagination. The nature of mathematics is discussed as based on this framework from intuitionist and logicist perspectives. Logic defines the structure of space and time.

Author Bio

Darius Malys is an undergraduate student of Mathematics at the University of Glasgow, Scotland. His passion is philosophy of science, especially digital philosophy. He research Kant and German idealism in the context of the present day problems of physics. At the moment he works as a CCTV systems engineer to earn money for studies.

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Darius, thanks for your thought provoking essay, though I do not follow everything Kant said.

Kant is right that our experience is the root of our being. The external world is constrained to meet the requirements of our logical thought, but Kant wants more than that. He thinks temporal causality, space and time must be fundamental because they are how our mind perceives the world. Hume was skeptical of this and rightly so. Relativity and quantum theory have shown us that the intuitive instincts that are programmed into our mind are not the way the universe works. The only thing we can rely on to understand physics is logic. Our experience is one of the infinite set of logical possibilities that adhere to the rules of mathematical consistency. Those rules are prior, not our thoughts.

Einstein dealt a big blow to Kant and if space and time are emergent then much of his philosophy is wrong, but I think there is a deeper sense in which Kant had a good point to make about the realtionship between our conciousness experience and nature. If we are passengers on a ship then at least we can say that the ship had to be built so that it could carry passengers.

    ``Relativity and quantum theory have shown us that the intuitive instincts that are programmed into our mind are not the way the universe works. The only thing we can rely on to understand physics is logic. Our experience is one of the infinite set of logical possibilities that adhere to the rules of mathematical consistency. Those rules are prior, not our thoughts.''

    I disagree.

    Relativity and quantum theory have both had some success at predicting. However, neither seems able to agree with the other and neither seems able to be a model of the very big and the very small. Relativity has had some ad hoc modifications. Quantum theory still needs a physical basis for the double slit experiment and the ``which way'' experiment of Ashfar. Quantum also has many physical models (Copenhagen, Bohm, many worlds, etc.) that reduce to the math of quantum theory - not a good indication for a universe model. These are indications they have reached their limit. A new model is required. Therefore, they have not shown us how the universe works. Worse they have shown us that they are not the way forward. However, any new model must reduce to both of them in their respective spheres of influence.

    ``The only thing ...'' is false. We just haven't found it yet (but I and some in this forum have suggestions). How about intuition? We are born in this universe and have survived in this universe. To survive means our brains must be hardwired with some ability to know how the universe works. Our survival attests to this. That is, the rules shape our thoughts through evolution.

    The similarity of human behavior suggests that there is some common view of the rules of the universe - one other paper commented on the Indian Elephant. There is even the discussion of a created or eternal universe in religions - it seems religion was first in this discussion.

    Last year you commented that the scientific community is exerting great social pressure to stick to the status quo. This is still another reason for viXra. I'm a little surprised you defend Relativity and quantum theory.

    Dear Darius Malys,

    You have done a very nice job on your chosen topic. I tend to agree with Dedekind, Frege, and Russell that "mathematics is an extension of logic." Considering the time frame, ~1780, Kant constructed an amazing edifice, which in many regards still stands today. But naturally when it comes to physical details his is more metaphorical than actual. I would like to remark on just a few of the statements that caught my attention and interest.

    When Dedekind states "numbers are free creations of the human mind", I believe he has somewhat missed the boat, unless what he really means is "the conception of numbers...". Numbers appear at all levels of the universe, whether energy levels or number of electrons in an atom. Proteins and DNA "count", for example the initial length of telemeres, or the numbers of fingers and toes, or even the numbers of cat whiskers, those little immediate sensors that work in the dark and map into a matrix in the cat's mind. Kronecker said, "God made the integers, all else is the work of man." But integer numbers appear everywhere, and the physical means of producing integers are almost limitless. I briefly address this in my essay, which I invite you to read.

    I did very much like your quote "the most fundamental and original act of imagination is the production of time consciousness." There's food for thought!

    I also appreciated: "the principle of contradiction... being the universal and completely sufficient principle of all analytical knowledge." Very good!

    I don't know whether you are speaking metaphorically or literally when you say "different rates of time parameters create different rates of vibrations of cells. Our thoughts are waves in the structure." I would not take that too literally.

    Nor would I take too seriously "the framework is one quantum entangled network which forms the unity of consciousness." I've written earlier essays on consciousness and my current essay reflects on entanglement in a new way.

    I've picked out just a few of the many interesting things you cover, but I think you've done a masterful job in your treatment of this necessarily complex issue. As you say at the beginning, a Theory of Everything should synthesize philosophy, cognitive science, natural language, information theory, mathematics and physics into a single framework or system of our reason itself. My belief is that this will not be possible until we have a much better understanding of physics. Our current theories do not hold together and cannot possibly be true. So until this occurs we can only go so deeply into a theory of everything.

    I very much agree with your summary: "It is not mathematics which is fundamental. It is logic."

    Good luck in the contest.

    My best regards,

    Edwin Eugene Klingman

    Thank you for your comments.

    If you found this essay interesting you might find my full project interesting as well: https://www.academia.edu/8991727/Phenomenal_World_as_an_Output_of_Cognitive_Quantum_Grid_Theory_of_Everything_using_Leibniz_Kant_and_German_Idealism

    I have been thinking about possible Theory of Everything for 6 years now, and this is my humble result.

    My starting point is this: if I want to understand everything (and find a ToE), first I need to understand how I understand things in general. How much can I understand? Why? How science is possible? It is our reason which understand things and forms theories, thus I must understand reason itself. I must look for universal epistemology and derive a ToE from there. I must look for a framework of all our possible knowledge.

    -----------------------

    I do not agree with everything Kant said either. However, in this essay I mostly relied on him because of the lenght limit. I used Kant to derive an ontology (the fundamental structure of the mind according to which our thinking and experience is organized) from which, I argue, it possible to derive universal epistemology. This project is greatly unfinished, it just lays a framework and directs further thinking. I do not have any funding and do this as a hobby, so I have to leave it until I get a PhD or get a decent funding so I don't have to work outside my studies.

    As an undergraduate student of maths and physics I always wondered what is the relationship between maths and physics? For me it sounded inconsistent and foolish to study maths and not know how it relates to the world around you! I liked Plato, Leibniz, Spinoza, Descartes but they did not give me an answer. Pythagoras said that everything is numbers. So what? This is not a decent argument. When I started to read and understand Kant's ''Critique of Pure Reason'', for a couple of months I thought it was the best book ever written. It really answers why our perceived Universe is described by maths and why ''a doctrine of nature will contain only so much science proper as there is mathematics

    in it''. Of course, Kant gave just the very basics. I know I must study such thinkers as Frege and Russell. But I think that the most important thing is to study them in the context of this structure: http://upload.wikimedia.org/wikipedia/commons/0/08/Flower-of-Life-small.svg .

    I know it sounds mystical or like pseudo-science, but my original insight is that this structure (the flower of life) is what Immanuel Kant calls the original synthetic unity of apperception. It is the fundamental structure of our mind. It is the structure where our thoughts are formed. Units cells are connected by logic, logic is fundamental. So it is an invariant structure underlying all our possible knowledge about the Universe and as such is an ontology for ToE. It is the framework of the cognitive faculty of understanding within which all our understanding about the Universe originates.

    I do not agree that Einstein refuted Kant's philosophy. There are many papers which argue that Kant's philosophy supports non-Euclidean geometry. My original insight that transcendentally ideal space (our cognitive framework which is 2D) is Euclidean, while empirically real space (3D) is non-Euclidean. In my essay you can find some relations to gravity as an entropic force. I believe gravity is an emergent phenomenon which arises from the difference in information-processing. As a unit cell vibrates, it processes information (performs synthesis which is the work of what Leibniz and Kant call spontaneity). So the information (energy) flow is distorted by different rates of information-processing and that in empirically real space and time appears as gravity.

    Kant's last major book was unfinished. It was called ''Transition from the Metaphysical Foundations of Natural Science to Physics''. At the end of 18th century Kant argued that there must be 4 fundamental forces of physics (surprise, surprise!) which must be derived from the laws of our thinking. I will investigate this further when I have time...

    All the best,

    Darius Malys

    There has in fact been a lot of progress towards combining relativity and quantum theory. True, the work is not complete. If it turns out that finishing requires us to return to concepts of absolute time and space with deterministic realism then Kant will be vidicated. But that is only the view of Kantian extremists and is very far from mainstream thought (you are still welcome to persue that direction if you wish)

    On the other hand there are plenty of insights provided by Kant that continue to fit well with current thinking. His idea that the observer participates is crucial to both relativity and quantum theory. These are Kantian ideas that I can embrace and that I think are relevant to the relationship between mathematics and physics. So your essay is appreciated from that perspective.

    By the way. I dont think my defense of relativity and quantum theory is in contradiction with encouraging and enabling people to challenge the status quo. As Smolin says, we need to be radical conservatives choosing what to accept and what to reject. We each need to test our own choices for how things work and the best way to do that is by letting those who disagree with us have their say. My own work also challenges the status quo, even if I think that relativity and quantum theory are correct.

    Dear Philip Gibbs,

    you say: ''He thinks temporal causality, space and time must be fundamental because they are how our mind perceives the world.''

    Kant distinguished between the reality as it is in itself (noumena) and the world as it appears to us (phenomena)!

    According to Kant, space, time and causality are NOT fundamental to the reality as it is in itself!!! Space and time are just pure forms of sensible intuition in which we receive sense-data (intuitions) and perceive phenomena. They are fundamental to the way our brain creates our experience of the world (phenomena). Causality is the product of our understanding (look at Kant's table of the categories and the second analogy of experience). The understanding prescribes laws to the appearances in space and time. But causality is not the property of things-in-themselves. Kant build his philosophy using Hume, Locke the rationalists. He agreed with Hume in some sense.

    you say: ''Relativity and quantum theory have shown us that the intuitive instincts that are programmed into our mind are not the way the universe works.''

    I don't see any problems with 20th and 21st century physics and Kant. Actually Kant's philosophy predicted modern physics and is compatible with it. Just look at ''the Postulates of Empirical Thought'' in the light of quantum physics. You probably hold space, time and causality to be the properties of things-in-themselves. That's completely wrong.

    Kant does not hold Newtonian view on space and time. In many cases he goes against Newton.

    ''If it turns out that finishing requires us to return to concepts of absolute time and space with deterministic realism then Kant will be vidicated.''

    It seems you have never read Kant. First of all, according to Kant space and time are NOT CONCEPTS but intuitions (original German word for that is Anschauung).

    12 days later

    Darius,

    Thank you for providing an interesting essay. I have only one concern regarding the suggestion that logic is the fundamental aspect of nature and not mathematics. It would appear we need mathematics to define a statement of logical. Therefore, it would be logical to state mathematics is more fundamental than logic since it is a derivative of logic. It is almost a "chicken and egg" comment, but we must compare a statement in an argument which in effect is an equation whose validity can be proven through comparison or logic. I am afraid it is logic that branches from mathematics.

    In any case, thanks for the good read!

    Best Regards,

    D.C. Adams

      hi,

      thanks for your comment. If mathematics is fundamental and logic is derived from mathematics, then what is the place of natural language in the hierarchy? Is natural language somehow less fundamental than maths, is it derived from maths??

      6 days later

      I think your remark is the same as asking what is the purpose of doing metaphysics, ontology and epistemology? You can study nature bit by bit, as an animal tries to investigate his environment. But you can be smarter and ask questions such as: what is the nature of my experience of the world? why can I understand nature? how much can I understand in nature? how can I understand things in nature? Only humans (self-conscious beings) can ask such questions, and only humans can understand nature completely. You can build a systematic epistemology, a metaphysics from which you can derive your theories. All our theories come from our brain, from our mind. So if we want to understand nature completely and form an ultimate theory we must understand how our mind understands things in general. It is our brain which generates our experience of the world and it is our mind which forms theories about the world such as relativity, gravity and black holes... I was looking for a universal framework underlying all our theories.

      in addition to that, if, as you say, logic is derived from maths, then I don't see how to derive fundamental forces of physics from that.

      in other words, I was looking for a fundamental structure of the mind and discussing how mathematics relates to it. In my full project I also discuss how physics and natural language relates to it.

      The Leech lattice forms a large error correction code, which can operate on the quantum level. This could form the basis for the ultimate encryption/decryption system. If we should have contact from extraterrestrial intelligence it could be that we decipher their messages using this. If so this would gives some weight to the idea that how our minds operate, or math-linguistics of human thought, is tied to a universal structure. The Leech lattice is a spatial subgroup for a theory of the 26 dimensional bosonic string.

      LC

      5 days later

      Hello Darius,

      I enjoyed the excursion into Kant's world. There are a couple of areas where my model shares similarities, such as building up our framework from time and space, and for the unit to have binary/ contradictory states. We differ in that while you prefer the unit circle, I prefer the extended point (Pythagorean monad) as the unit.

      Having said that, may I ask if your Unit circle can perish or is it an eternally existing thing?

      May I also ask that when you say for example, 7 + 5 = 12 is true, that is to say when you put 7 apples in a basket and add another 5 apples, you get 12 apples; that the truthfulness of your equation and mathematical statement contains an unstated and taken for granted assumption that none of the apples can perish. But suppose, things can perish. Then, if you put 7 apples in a basket, and later add 5, then it is possible that when you count to find the sum 7 + 5 тЙа 12. My essay is partly about this possibility. You may want to take a look.

      All the best in the competition,

      Akinbo

        5 days later

        Hello,

        Pythagorean monad is similar to my monad (unit cell). A point expresses pure consciousness (or pure Reason) which is outside space and time and is eternal. When pure consciousness starts to think it acquires a boundary, that is becomes a unit circle. Time parameter moves around the unit circle which creates the vibration of the unit cell. This vibration (spontaneity - motion of the time parameter) is the process of thinking.

        In the beginning 1 circle appears. Dialectic is the fundamental law of our reason. To be self-conscious it needs something other - the 2 circle appears and limits the first. They vibrate at opposite phases. That's the Big-Bang.

        I will take a look at your essay when I have time.

        7 5 = 12 is true because it follows from the fundamental structure of our mind which is at the same time the fundamental structure of space and time. But it is true only for objects in space and time.

        Personally, I may not attribute the property of consciousness to a fundamental entity like a 'point' or 'monad'. I believe consciousness is a property of composite things.

        Then in saying, "7 5 = 12 is true", the assumption is that during the mathematical procedure of summation, none of the things to be summed up can perish. This is only very highly probable but not a certainty particularly on the quantum scale. The probability of not perishing during summation is not the same for the classical and quantum scale. If one electron perishes in an apple, it remains an apple that can be counted. But when you go further down you may get virtual and other particles that can pop in and out of existence during counting.

        Anyway, that is what I think. May be wrong.

        Regards,

        Akinbo

        Darius,

        I did enjoy reading your essay although I lack a deep understanding of the philosophical references in the essay.

        I view the search for understanding as a process involving all the tools we have at our disposal including, logic, mathematical models, scientific observations and descriptive models.

        I think your analysis goes deeper that this in a search for meaning.

        I think your approach to a theory of everything is too ambitious: You say a Theory of Everything should synthesize philosophy, cognitive science, natural language, information theory, mathematics and physics into a single framework or system of reason itself.

        I think that some of the above are more easily dealt with separately as disciplines in themselves and a Theory of Everything in physics is attainable in isolation.

        This Theory of Everything in physics would be a complete and comprehensive description of the physical universe with a description of properties and a set of mathematical equations that model the physics.

        The philosophical underpinning of the logic of the physics can be dealt with as a separate exercise.

        Regards

        Richard Lewis