Life is so incomplete

This post won’t be as melancholy as the title may suggest; it forms part of my quest to acquire the basic ingredients of human existence. Mathematics is the unspoken language of nature, in which we must strive to achieve fluency if we are to bask in the richness of the universe in which we are marooned. When we speak in our mother tongue we tend not to concern ourselves with proof. We may perceive a distinction between a “lie” and the “truth” but we allow for expression of opinion, without which the world would be much more honest but considerably less interesting. Take the statement I can’t go to drinks this evening, I am really busy. How taken aback you would be if the response to your statement was; prove it? Conversely, if I came to you with a statement for finding the nth prime number in the language of mathematics, how taken aback would I be if the response didn’t come; prove it?  Is it that we just don’t require rigorous proof in the spoken language, or are we just accepting a set of truths deeming them not to require any derivation? There are parallels between the mathematical language and spoken word; the most important being underlying conventions or axioms (rules taken as true), which in our own human way we take for granted. Remove these rules – chaos reigns.

Kurt Friedrich Gödel was a mathematician, logician and philosopher and widely regarded as one of the most important figures in mathematical logic (and certainly the most important contemporary). At the tender age of 25, Gödel was able to prove his incompleteness theorems, the results of which have far reaching impact. I really should try and pack a little more into my years. His theorems can be summarized as;

  1. Within any system where enough arithmetic can be carried out there are statements written in the language of this system that can’t be proved or disproved.
  2. A system cannot prove that the system itself is consistent, assuming it is in the first place (which we could never prove!).

Roughly speaking a “system” means a set of rules within which we can create new theorems. We generally call these rules axioms; they are the truths of the system if you like. We then have operations we may perform within our system, which coupled with the axioms allow us to determine for any formula (in the language of the system) if it is a proper derivation or proof within the system. The natural numbers are a very basic example, coupled with the arithmetic operators we know so well. If the above definition didn’t come naturally it might be worth a reread; it is quite important. Now a system can be described as complete if for any statement we generate within the system we can prove/derive the statement (within the system). A system is consistent if, for every provable statement I cannot also correctly prove the contra. To use an example; if I can prove I am a man, for the system to be consistent I better not be able to also prove I am a woman; which, to my knowledge I cannot.

So what does all of this mean?  It is really important to understand that a statement being false is not analogous with a disprovable statement. Although not a perfect example, consider the idea that no number of white swans may prove the statement that all swans are white – all swans might be white, but even with a large army of white swans I am nothing more than a man with lots of swans. In order to get your head around this we think about computer programming; if you read about Gödel’s theorems this is what you will often see cited.

We use a “perfect” computer where we don’t worry about petty things like processors, electricity and alike. Just one big super computer that exists (in your mind). The computer can perform the usual arithmetic operations of addition, subtraction, division and multiplication. You can program your computer to perform tasks, you really can pick anything you like it’s your computer. You might be interested in knowing if a value is larger than 100 – well tell the computer to subtract 100 from the value and if the result is 0 or less display false. We can get more sophisticated and program our wonderful computer to determine primes (a sequence of divisions), Fibonacci numbers or square numbers. We can do whatever we like with our supercomputer, right? Wrong.

As a direct consequence of Gödel’s theorems, there are statements (only involving arithmetic, we aren’t asking the meaning of life), correctly inputted into our not so super computer which it cannot prove or disprove. The computer genuinely cannot decide if the outcome is true or false; nothing worse than an indecisive computer. To make things spookier, say you introduce a new axiom to the computer, a cheap work around to say that the bothersome statement is either true or false (it may well be intuitive) things don’t get better. You would either generate more statements the computer cannot decide between (1) or you would create results that contradict the pre-improved computer which are valid in the new post-improved computer – inconsistent (2). When this whole math-bomb dropped people hoped it was just a quirk of the system as defined rather than the reality of systems we know and love. Alan Turing, among others showed that this is reality. An incomplete or inconsistent mathematical meltdown keeping logicians up at night pondering all they once knew.

The theory has been used in a huge number of areas; including as an attempt to prove the existence of God I know some of my readers will be delighted to know! But alas, that is not the direction I intend to take for the swansong; instead I delve inside your mind which is much more interesting. I exaggerate a little, we will all need to journey deep inside our own minds. If we  consider our brains as essentially machines (your choice), then do Gödel’s theorems apply? We hope – dearly – that our brains are consistent (although I do sometimes find myself questioning this watching political commentary today). So we arrive at the conclusion, based on the above, that there are basic statements with arithmetic operations we cannot prove or disprove within our own brain – which would be the strongest line of evidence that we cannot ever construct or program a human mind. Rodger Penrose is a firm believer of this, who controversially states that a human mind is capable of knowing the truth about these Gödel-disprovable statements; this form of intelligence is never able to be computed. More on this in the future, but I should tell you this has been met with some severe criticism; centrally that we cannot know a human brain is consistent without much more understanding of the inner workings, especially due to the number of mistakes we make we could very plausibly be neurologically inconsistent, (and perhaps some more than others!).

That aside, Gödel’s theorems raise some profound questions around the foundations of mathematics and the nature of our brains. The mathematical proof is quite dense and I do not intend to outline it; if you are interested I actually find Wikipedia the most usable but please let me know if you have more user-friendly sources.  The language of logic is a difficult beast to tame.

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71 responses to “Life is so incomplete

  1. My relationship to mathematics is, unfortunately, rather incompatible. I have always had a bad memory and my struggle to retain the basic multiplication tables was a very tough grind for me as a kid and although I have passed courses all the way up through integral calculus I always felt on insecure grounds on the whole business. My understanding about math is that it involved systems that were consistent but that did not mean that they necessarily related to reality. Even Einstein said “As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.” And many of the more far out mathematically derived physical theories such as string theory have yet to be observed in physical research. I do not depreciate the importance of math in science but it is not a tool I feel comfortable with.

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    • In my personal interest, theoretical physics mathematics and physics are so intertwined you can easily study postgraduate having taken either degree – often maths is more favorable! The point about mathematics, particularly when we are applying it to the real world is it is a modelling tool which can give frighteningly accurate results – provided the model is strong. Whilst on the one hand mathematics leads us down roads we have no experimental evidence for; let us not forget that mathematics gave us quantum mechanics – an otherwise impenetrable fortress. Maths should never be about memory – in fact many courses are now taught with a handbook you can take into the exam. It is all about modelling reality, or probing the unknown – something that I think is just wonderful!

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      • I tried to become an engineer at CCNY back in the early 1940’s and got through the preliminary physics, chemistry, and biology courses and the math through integral calculus but these were much like the science courses I had taken in Stuyvesant High so there was no problem but my studies were interrupted when I joined the Air Force in 1944 in WWII and the atomic bomb made me want to become a nuclear physicist and that’s when it all fell apart because of my math allergy. I tried to get a grasp of theoretical math analysis where the text was “What is Mathematics” by Courant and Robbins and it was fascinating but I couldn’t grasp the math even though the instructor and another student tried to rescue me – but it was no use. It was a wonderful experience but my mind simply refused to digest what was necessary. I have no doubts math is a wonderful mental dimension but I simply have not the internal mechanisms to accept it.

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      • What an interesting background you have! I refuse however that you do not have the mental dimensions to cope with it – I firmly believe you do, I think it’s just about finding your own way of understanding it. In addition, pure maths is an acquired taste. I love it; but applied maths is much more usable and rewarding if you are looking to study “real world” problems; and because it so easily maps to real world problems you will find it much easier to comprehend!

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      • In my conflict with math I discovered there were two basic approaches to the multiple disciplines involved. The prime approach is to appreciate and deeply understand all the integral concepts behind each minor discipline such as that monstrous formula for solving quadratics and the very strange methods for dealing with determinants and the blind acceptance in basic calculus of swallowing that reducing distances to infinitesimals to be summed resulted in the simple ability to determine the precise lengths of curved lines loses me. The formulas were relatively simple to memorize and apply but the deep understanding of juggling infinitesimals was a dark area that I could not intellectually penetrate. This depth of comprehension is a necessary mental function that was the true understanding of mathematics and it was forbidden territory to my mind. The rote application of formulas and procedures was always taught and is the procedure of a computer’s algorithms that does not require a conscious mind, something I have been cursed with from birth. The very confrontation of the infinite and the fusing of infinitesimals boggles my mind and leaves me confused.

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      • I know what you mean, although often in my experience it actually is only once I have churned through the formulas a few times that I actually come to appreciate the underlying subtlety…. I don’t think you should give up on it! It is one of those things that one day just clicks and after that it is very difficult to see why you didnt understand it, or indeed explain it to others!

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      • Let me outline one simple difficulty I have with mathematics. It derives from the prime difference between mathematical geometry and the real world. In math lines are continuous and no matter how short an infinitesimal line may be it still contains an infinite number of points. The concept of a line (or any other geometric form) differs completely from its actuality in its property of continuity. A real drawn line, with the proper magnification, shows discontinuities in distances between atoms and subatomic particles. Infinitesimalability is a fantasy.

        Take a right triangle with one side horizontal and one vertical. To join the ends forms a hypotenuse and the hypotenuse is shorter than the combined length of the horizontal and the vertical. Instead of drawing a hypotenuse, if a short vertical is drawn from the open horizontal end and at the end of that vertical a short horizontal is drawn towards the vertical of the triangle and this process is continued creating a series of steps eventually those steps will reach the upper end of the large vertical and if they are measured properly the created zig-zag of the short lengths will approximate a hypotenuse except that the total length of all those short segments will be equal to the sum of the lengths of the original long horizontal and vertical lines. No matter how you shorten the lengths of the pseudo-hypotenuse (and, of course increase their number to connect the open ends of the large vertical and horizontal, the sum of all those tiny bits remains the same. And even if they are whatever a maximum of infinitesimability is attained they will never become the length of a true hypotenuse. This is my problem with the sum of infinitesimals. What magic transforms those infinitesimals into a straight line? Perhaps I am stupid but this is my problem with calculus.

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      • You are not stupid at all – a very analytical mindset and an interesting point. I think the devil is in details; essentially when we are looking at things like right angled triangles they are considered to be constructed with model lines. We simply ignore all of the problems you mentioned and assume the line to only have dimension in the x and y dimensions and not have any internal or external roughness. Why? Because that assumption works when you are dealing with centimeters or meters. Millimeters are fine too. But indeed when you get onto the scales you speak of a line cannot exist as a line because there is nothing small enough to make it! Fortunately we don’t have too much need to draw right angled triangles in quantum distances; but as you rightfully point out mathematics does seem to break on these length scales.

        Your zig-zag example is a nice one; the answer actually is very similar to my Achilles and the Tortoise problem; that is when the sum over infinity sums to a real number. It is simply a value upon which the value converges as we sum up. But don’t think about infinity think about a margin of error that is more human. Maybe you can only measure to 0.1 of precision. So you keep going until you are at a point where you have agreement to that accuracy, and the world is a little clearer! Summing to infinity is loosely analogous with this; although there are not direct parallels with this finite process.

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    • It probably boils down to my fundamental sense which which is a construction of an identity made by my central nervous system to navigate through the artificial reality constructed by my mysterious CNS out of the nerve impulses from my sense apparatus. In this manufactured construction the abstracts are fashioned into a “reality” where all lines are continuous and all planes flat and Euclid reigns. It is the shadow world of Socrates and turning my head to face real reality is not possible.

      CREATION OF THE UNIVERSE

      Sits in my skull
      A curled gray beast
      Molded to its cup of bone,
      Alone.

      The outside world
      It cannot know.
      Not light, air, bulk nor hues.
      Just clues.

      The nexus of
      A finespun net
      Which terminates its axon knout
      In doubt.

      Its billion lines
      Transmit responses
      Sifting pulses; all compiled
      And filed.

      Confusion, first ubiquitous.
      Unvectored bits, zero, one
      ‘Til the sources are assigned,
      Aligned.

      Woven nerves
      Festooned with figures;
      Puzzled with the patterns, matching,
      Hatching.

      Lacing through
      From point to point.
      Architecting, congruencing,
      Sensing.

      Congealing concepts.
      Counting, seeking.
      Logic engine freely dreaming,
      Scheming.

      Fitting this,
      Forming that,
      Smoothing, joining, multiplying.
      Trying.

      Granting trope
      Its own dynamic.
      Now it all agglomerates
      And mates.

      Sloughing off
      All errata,
      Chaos clears. Universe
      Appears.

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  2. Incompleteness is facilitates blossoming of character. Through ambivalence and divided passion, we learn to expand or delineate our reality into something more beautiful, something more complex, something that does justice to the magnificent chaos that encompasses all of us. There may be no undulating, ubiquitous higher order of existence that the supercomputer activists opine. Instead, things may be an oscillating, topographical work of art that we gaze into again and again and again until we reach forever.

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    • I don’t think I will ever be able to accept a reality we cannot rationalise.. I will certainly need to change the site title if I do! An interesting perspective none the less, you are correct it is entirely possible the truth is beyond human comprehension; I just ignore that or to be honest I might not bother getting up in the morning

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    • Aside from the unfortunate reality that truth, like forever, is not a static goal but a handy tool like those straws chimps use to fish out ants or the twigs crows use to reach bits of food unreachable by beaks alone.
      I have lived long enough to see the outer universe modified remarkably and truth, like those handy straws, transmogrifies immensely over time. Our unfortunate computer pets set to graze on esoteric algorithms and are rarely freed to leap through nature and speculate on the nature of reality. Naturally, like young babies, they stumble and fall and have little internal wisdom enough to right themselves and climb up a tree, a heritage we monkeys delight in with hardly a thought. Give them a while to work out the proper strategies and they may become as charming as dogs.

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  3. You point out correctly that Goedel’s theorem is restricted to a fairly narrow problem: proving that a proof system is “correct” – i.e. – that it’s axioms and operations are consistent. In other words, we can’t take a set of axioms and apply the operations to disprove any other axiom.

    This seems to lead to the conclusion that we can’t trust our proofs of anything, which means that there are no guarantees that our expectations will be met. Unfortunately, expectations are undermined by many other problems, among them determination of initial conditions, noise and adaptation. The last is the special bete noir of sociology, as people will often violate social norms in order to assuage primitive drives.

    At this point in my life, I am actually not at all troubled by these problems. Satisfaction is not found in knowing the truth, it is found in realizing creative possibilities. If we could use mathematics to optimize the outcome of social and economic systems, we would have no choices left. Life would become terribly boring. So what is interesting to me is to apply understanding of the world to imagine new possibilities. Mathematics is a useful tool in that process, particularly when dealing with dumb matter.

    This brings me back to the beginning of the post: you state that “mathematics is the unspoken language of nature.” If there is anything that Goedel’s theorem disproves, it is precisely that statement. Mathematics is a tool, just as poetry and music are tools. At times, both of the latter have transported my mind to unseen vistas; mathematics has never had that effect.

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    • You raise a very interesting point; if we could optimise everything then would we take all of the joy out of being…. you may well be right. I know I get a lot of my satisfaction from the quest to know more. Although I disagree that Godel’s theorems disprove my original statement in this sense; language is essentially about describing things. That is why you can have different languages but they are easily translatable…. bread/pan/brot etc…. we all know what they mean because they all describe the same thing. In exactly the same way, mathematics describes things that actually exist; that isn’t to say nature is mathematics at all – mathematics is the language of nature but it is just as human in its construction as the spoken word. But is matter not matter because a human invented the label? Matter is matter.

      To be, these theorems don’t break down all of our proofs; but what they do show is a vital point about logic. One which I think is going to become and increasingly big issue as the quest to understand and build artificial intelligence increases – can we every build a mind as intelligent as a humans when a human can know the answer to a non-programmable result? We hope so! Or rather I do – I do appreciate it’s not for everyone

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      • I appreciate your enthusiasm, but I must caution that the mathematical analogies in classical physics cannot be extended in the same way to the quantum realm. Richard Feynman warned us that there is no coherent philosophy of quantum mechanics – it is just a mathematical formulation that produces accurate predictions. Ascribing physical analogies to the elements of the formulation has always caused confusion. An extreme example was found in the procedure of renormalization, in which observable physical properties such as mass and charge are produced as the finite ratio of divergent integrals.

        Regarding human and digital intelligence: one of the desirable characteristics of digital electronics is its determinism. The behavior of transistor gates is rigidly predictable, as is the timing of clock signals that controls that propagation of signals through logic arrays. This makes the technology a powerful tool to us in implementing our intentions.

        But true creativity does not arise from personal control, which only makes me loom bigger in the horizon of others’ lives, threatening (as the internet troll or Facebook post-oholic) to erase their sense of self. Rather, creativity in its deepest sense arises in relation, in the consensual intermingling of my uniqueness with the uniqueness of others.

        Is that “intelligence?” Perhaps not – the concept itself is difficult to define, and I believe that it arises as a synthesis of more primitive mental capacities, just as consciousness does. But I doubt very much that Artificial Intelligence is capable of manifestations of creativity, because fundamentally it has no desires. It is a made thing, not a thing that has evolved out of a struggle, spanning billions of years, for realization. Our creativity arises out of factors over which we have no control: meeting a spouse-to-be, witnessing an accident, or suffering a debilitating disease. We have complex and subtle biochemical feedback systems which evolved to recognize and adjust to the opportunities and imperatives of living. We are a long way from being able to recreate that subtlety in digital form, and without those signals, meaningful relation cannot evolve, and thus creativity is still-born.

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      • Of course they cannot be extended in the same way, that is indeed where my deep fascination with quantum mechanics comes from; but that isn’t to say there isn’t a fully coherent philosophy – we just don’t understand it yet. There is so much wonderful and worthwhile things to know about the area, at this point in time it is my most probable area of specialization. I have recently signed up to some advanced QM courses. Let us never forget with computing that quantum computing is the future!

        Can AI ever be creative in the way that we see it. You are right we are a long way off being able to create it but the question that interests me is is it even possible. My view point is that under the right conditions we can – it just seems impossible to me that something I fundamentally believe to be a physical property can never be reproduced but it remains to be seen. In my most recent meet with the Institute of Physics over advances in AI we are looking at reactions and event planning so some way to go indeed! Hawking gave a very interesting talk on this yesterday actually there was a live stream from Cambridge University

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      • As you are considering this as an area of specialization, another caution: when I was reviewing quantum computing, most of the theoretical work had a fundamental defect. The authors assumed that the operating machine could be organized in logic gates at which the particle state was known to have collapsed. This was convenient for them in that it made tractable the calculation of the Hamiltonian of these complex system. However, it violated the formalism of the theory.

        I am afraid that you may find that the lack of a coherent philosophy of quantum mechanics fosters similar sorts of imprecision in other applications. That has certainly been my experience.

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  4. Pingback: Intelligence and Creativity | everdeepening·

  5. Pingback: I know some who would very much find this interesting. I hope they see it. Life is so incomplete — Rationalising The Universe – The Frantic Poet·

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  7. I enjoyed your exposition of Godel it’s a difficult concept for us ordinary mortals. I notice that Prof Steven Hawking wrestled with its application to physics in ‘ Godel and the End of Physics’ which caused quite a stir. I struggle through it getting a glimpse here and there.
    Alon Amit seems furious in his refutation of Hawkings use of Godel. Once again a tough nut for me but I notice at the end he said: we may not find an ultimate theory because we are not smart enough!
    There is a feeling today anyone can do anything which is nonsense. My IQ is about 105 and I know I have limited understanding that is why I seek out the experts to get an idea of what is going on.
    Steven Pinker in his tricky but interesting book ‘ How the Mind Works ‘ speculates at the end that the evolutionary purpose of the human brain was survival and we may not be equipped to solve many of the problems we seek to understand.
    He tackles the Wallace Paradox and thinks he has solved it but I’m not so sure. Alfred Wallace believed the Mind of man could not have evolved by natural selection since cave man did not need such a brain to survive. The Brilliant Wallace went off the rails and became a spiritualist.
    You say the language of nature is mathematics but I think nature has many other languages , how mathematics explain the piano sonatas of Beethoven or the works of Chopin? Indeed why are we not all so gifted and where does the gift come from? Was Darwin using mathematics in his search for the origin of species?
    Now we know that computers can now out play any human at chess but they do it by number crunching not by judgement. So my question is how do humans play since we cannot number crunch ? and the answer is judgement whatever that may be.

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    • Hello and thank you for your comment! You raise some very interesting points. Whilst I do generally agree with you that the actual problem solving of the big issues in Physics are going to fall to an intellectual elite I do believe everyone should be able to engage in them. In the same way that art is produced by a small number of individuals, but many visit art galleries and enjoy the work or relax in their free time drawing and painting. It’s no different, I think it is vital everyone is engaged and involved, or at least as many as are willing to listen.

      I don’t believe we are able to pose these questions we are unable to solve. I think that is the chatter of when we are in a rut, soon to be followed by a period of relentless progress.

      I do agree with you mathematics is unlikely to be the only language of nature – but rather our own language which we constantly evolve as we need. Just like the spoken word it’s a description too; when new things to describe emerge we must make allowances within our language. I do believe you have however touched nicely on the lead in to my next post; in the final paragraph I think you highlight the difference between being alive and being conscious. Currently we seem to be able to do things we are unable to program – the basis of Penrose’s argument; so are we to deduce it is unprogrammable or we don’t yet know how to program. I sincerely hope the later!

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    • It certainly does make you feel a little more special when you consider the absoloute struggle to comprehend how we perform some of the most basic of tasks

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  8. As I’ve mentioned, there is a major problem I’ve had with mathematics and since math is a universal tool in scientific discipline which has fascinated me all my life I have been searching in this discussion for what this fundamental barrier might be. This is a guess but it seems to be valid. Through elementary school, high school, and college I have worked my way through the many problems presented in the curriculum, and in the many texts because they were many steps in learning to be capable in facing problems, from arithmetic, algebra from elementary to advanced, from geometry, and analytical geometry and trigonometry and through the two basic calculus courses. Admittedly I had never been excited or devoted to mere problem solving per se nor understood how these exercises fitted into my life which has pretty much been never. As someone who worked with drawing and the arts and eventually gained professional status as a draftsman, the implements granted by geometry and even, to a minor extent, analytical geometry gave me useful tools for my work but even the elementary algebraics were mere academic puzzles to gain a grade. To put it in a way personal, I never used most of the mathematic skills, those skills used me to get through the damned courses. In any life situation I never found solving equations of multiple unknowns of any use at all. Perhaps the people today who work in digital areas or finally get into exploratory science need and use these mathematical tools but for me as just an ordinary curious guy who likes to work in wood or metal or plastics the math. Like those dates we had to memorize in history with no deep understanding of the deep economic or cultural forces that drove human history, were quickly forgotten once the grades for the courses were granted. How Many kids who get through math courses actually find that all those tricks with numbers played any role in living their lives? Very little for me and it was mostly a total waste of time. And best forgotten as a damned nuisance.

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  9. At the risk of becoming a nuisance over my personal confrontation with mathematics I was prompted to investigate what might be a neuro-physiological difficulty at https://en.wikipedia.org/wiki/Dyscalculia#Persistence which seems to relate in many ways with what has been bothering me. At the age of 90 it seems a futile effort to correct something that might have massively changed my life if it had been remedied when I had the bulk of my education in front of me but now is a mere curiosity in my development.

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    • Curiosity in personal development is the way in which everyone should learn – be it at 10, 50 or 90 – I don’t learn to achieve a goal as such. There i things I would like to find out for sure, but the satisfaction is the process not the results

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  10. Thank you for publicising Godel’s Theorem, which has always seemed to me to be highly significant. It shows the limitations of the mathematical models that we construct to describe reality. And because these models are so pleasing and beautiful, we can even confuse them with reality itself. It is strangely pleasing that mathematics here shows its own limitations in the ultimately impossible project of trying to fully model the real.

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    • Thank you very much for your kind comment! It really does raise some profound questions about the basis of reality within mathematics. There is no use blindly applying a tool in which you have no idea of the limitations

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    • Although there is little doubt that mathematics has its limitations my viewpoint as to reality itself is concerned gives me no assurance that the pitiful fragments out of personal and instrumental input that we integrate as someone fits together possible pieces of a jigsaw puzzle that combine in all sorts of strange ways will ever settle into one picture. Through time and experience our flat world curled into floating spheres and time itself has become an elusive spectre.

      RETREAT
      They spoke once of
      The broken edge
      Where world and sky
      Made meeting
      In catastrophe.
      Where seas fell down
      In steady roar
      Into the sky,
      Or pits of Hell.
      What happened there
      No one could tell.
      No one had seen
      Or cared to see
      This horrific mystery.
      When Magellan
      Sought to find
      This birthplace of infinity,
      The Earth had sealed
      Unto itself.
      Grand horror fell
      Back into the mind.

      Once there were
      Great man-shaped things
      That lit the stars
      And ate the moon
      And rolled the Sun
      Across the sky.
      They shook the earth
      And pissed the rain
      And laughed with thunder
      And disdain
      At mankind’s loss
      And silly gain.
      They told when
      To plant and sing
      And fear and die
      And everything.
      But, somehow,
      Upon looking close
      They proved far
      Too bellicose.
      The rules are calmer now,
      It seems.
      They’ve tumbled back
      Into our dreams.

      One God, at times,
      Is still up there
      Behind the stars
      Somehow, somewhere.
      He fusses on morality
      And fiddles with
      Our destiny,
      But seems, most times,
      If will is free,
      Existing inconsistently.
      His eyes are red,
      His thoughts are tired.
      His beard as white as snow.
      The ovens in his antique Hell
      Are burning very low.
      The World, I fear,
      Will soon dismiss
      This Father of
      Immortal bliss.

      There is no longer
      Any spoor
      Of Moon creatures
      Of Cavour,
      And Mars has turned
      To rocky dust.
      Barsoom, it seems,
      Is a bust.
      And so the monsters
      File away.
      Locally
      They’ve had their day.

      But out beyond
      Centaurus arise
      The monsters
      With their death-ray eyes.
      There, around alien fires,
      The spooks and gods
      And monsters stalk.
      The gods strum softly
      On their lyres
      While things
      With twisty pseudopods
      Drip acid slime and talk
      In garbled yowls,
      Soprano howls,
      Of starships come
      All filled with men,
      That monsters reign
      Supreme again.

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  11. The problem with mathematics is the reducING everything (or almost) to simple equations. Whilst it helps us understanding a lot, on the other hand it leaves us with a simplified conceptual view that puts aside the richness of perceptions and details!

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  12. Pingback: Gödel’s theorem – I can't believe it!·

  13. Although my presence at this site may strike some as overbearing it is a discussion that has become rather tasty to my internal appetites for probing emotional reactions to mathematics as a powerful tool. One of the initial declarations was that mathematics is a language of nature and that thought strikes a chord. Nature as a whole is a rather intractable totality, a massive assault on perception from multiple angles that one cannot confront with simple singular precepts. But the one outstanding family relationship it has with mathematics is its total divorce from morality. It is a juggernaut of necessities responding to a rationality that has no relationship to those prime fundamental drives and coercions that make me a human animal with emotions of respect and love and astonishment and delight and the wonderful warmth of being alive amongst fellow living creatures which is so precious to me in so many ways. The mathematics of striking a Ping-Pong ball with a paddle can strongly relate to that massive meteor that demolished most of life on the planet at the age of the dinosaurs and the figures involved have nothing whatsoever to do with how deeply I feel about that event.

    The response can be “So what?” but there are implications to current and past societies that indicate real difficulties. When the mathematics of government and economics and business policy demand hundreds of thousands of people be deprived of jobs, of education, of food, and of life itself, mathematics does indeed bring up sharply the inevitability of a falling meteor. When it becomes necessary policy of a successful business to merchandise useless medicines or raise the prices of foodstuffs or sell horrifying military equipment or indulge in financial scamming for the success of the enterprise that entails relationships with the heartless nature of nature and its linguistic of mathematics.
    Our current society has very much a deep relationship to the necessities of mathematics to the point of becoming a social metastasizing cancer.

    But what does this mean? We sure as hell cannot abandon numbers or the wonders of precise relationships. You cannot place guilt on dynamite for being explosive or cyanide for being poisonous. But something is horribly wrong with society. But somehow, numbers, like radioactivity, have become powers we cannot control. Mathematics as a discipline, is certainly not evil in itself, just as nature has no morality of itself. But somehow it lacks the safety of necessary humanity and perhaps that is my difficulty in dealing with it.

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    • Not at all you are most welcome here – consider it a second home! Mathematics may not have morality but nor does nature let us not forget. Morality is a human construction; you cannot expect to find a distinction between good and evil in the language of nature when there is no natural distinction made. In a socieconomic sense; I agree mathematics is one of your tools for tackling the problem but the mathematically optimum result is not always the correct when you try to cost in the intangible social aspects. So as I see it, it isn’t mathematics itself that causes the problem; but more under use of other considerations. We can never control mathematics, because for the most part we never did and never will own it. All we can do is verbalize it in our own human way

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      • This connects up with my disdain for money which is the invasion of numbers into social interaction. My brother with a very high IQ of about 140 did well in math and became very successful in his business. I saw people around me, even as a kid, who spent lives miserable working to provide themselves and their family with necessities. The lucky ones found ways to earn a living at stuff they enjoyed. I did interesting work but never really was successful in getting more than an average salary. Somehow math and money are interlocked in social values not terribly beneficial except to a very few and the rest of civilization is doing very poorly indeed. As an artist I never could see why some work was monetarily valued immensely over other stuff that seemed equally as competent but basically disdained. Perhaps it is totally irrational but it seems to me that money and the math of finance is fundamentally destroying life on the planet and the strange inhumanness of evaluating people in the cold heartlessness of amounts of money disturbs me greatly.

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      • I think it’s just the unfortunate laws of supply and demand. Jobs anyone is capable of doing or everyone wants to do don’t pay well and jobs not many people can or want to do pay much better. It means it is rare to hit that sweet-spot where you are doing something you love and someone is willing to pay you handsomely for it. That said; not assigning too much value to money and ensuring work is not your reason for being upright will together shield you from the difficulties of any career.

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  14. There is a book review at http://www.newstatesman.com/culture/books/2016/10/has-physicist-found-key-reality which seems to me to be relevant to this discussion. It deals with the fundamental discomfort I feel with current basic speculations of science and reality.

    THE TABLENAUT
    A wayward ant meanders on my kitchen table
    To savor ghosts of meals that passed this way.
    An archaeologist of ketchup spots and gravy stains
    To read the hieroglyphics of pate.
    A great explorer is this little guy
    To cross the huge expanse of asphalt tile
    And climb the table leg to wooden sky,
    Investigate eternal mysteries of kitchen,
    Surmount a sugar bowl to answer why
    And how the world is made and when, and who
    Could fabricate infinity with staples
    And wood and nails and paint and glue.

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  15. Considering this in a very simplistic way…perhaps we could deduce that the human brain does not belong to a system thus being unable to disprove itself. Perhaps the reason why we consistently develop such nonsensical explanations is because the truth is irreconcilable with the all the systems we inhabit

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    • I don’t think it is possible for something not to belong to a system without getting into realms of the unscientific. I could support and explanation that is is possible we will never be able to understand due to the complexity of the system – but I don’t really like to dwell on this too much as I feel it does not make for terribly productive thought processes!

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      • Although I have reiterated mercilessly my submission before the massively impressive conquests of mathematics that have won conquests after conquests in conflicts with the intricacies of the merciless universe I stand before the monument in somewhat the same awe as when viewing the marvelous creations of Brancusi who has removed the ravenous ferocity from a living fish to leave a most lovely
        massive flake of white marble in what might be seen as a solid melody in space. It is a triumph that shatters me to helplessness.

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      • I think this must be the most poetic comment on the site! I am sure you will eventually fall into a deep and loving relationship with Mathematics

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      • Marilyn Monroe and Fidel Castro and the queen of England I were born in the same year and like a hefty portion of the human race I also fell in love with Marilyn and I always wondered if I would see her in New York’s Central Park where she is said to have sat on a park bench. But whatever the level of my passion, even if she would be accommodating, I doubt my capability or courage to be able to consummate any deep relationship. This is very like my relationship to mathematics. I struggled through all the lower levels and made something of a foreplay with calculus and analytical math but I proved quite neutered. If math is a she, she found other sources to invite to bed while I sat on her doorstep with a bouquet of withered daisies. I stumble into poetry or vice-versa out of an ineptitude with common language since words, like numbers, are highly purified abstractions that mate in the ways of colonies of odd arthropods and produce progeny that hum with unknown threats and eyes that stare in yellow alien depths from the center of black holes. I have tried to feed these strange creatures with my chocolate covered marshmallows but their appetites hunger for the raw meat of the cosmos.

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