Dear Christophe Tournayre

Thank you for your comments. I agree with you that the essay raises those questions.

Three reasons for not calculating with ants are. 1) Confining the essay to the relationship between physics and math as artifacts of collective problem solving aims at thematic consistency and relevance to the contest theme. 2) A short essay is less work to read. 3) I do not know of any studies that measure the mean path length for ants to transmit information, or what might be measurable ant colony collective problem solving. Can it be done? Theory predicts a mean path length of ( 4/3 ) e, for the e the natural log for an ant network receiving information.

There have been attempts, notably around 1996 by the American Psychological Association (Intelligence: Knowns and Unknowns) to define what is measured by IQ tests (eg, "ability to understand complex ideas" etc) but I don't know if there is a consensus. I was forced to consider the rate of problem solving in 2005-2009 with my hypothesis that average IQs increase due to improving ideas. I agree that a rate of problem solving definition opens questions. For non-life forms, I would characterize the situation as: what principle common to non-life forms and human problem solving makes some natural phenomena resemble intelligent computation (your terminology)?

Regards

Bob Shour

Dear Joe Fisher

Perhaps some math and physics is more complicated than necessary and so seems incomprehensible because we have not yet figured out nature's simpler approach. This Einstein quote seems similar to your point: "as far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

On your wording correction: I agree that it is important to keep in mind that mathematics deals with idealized abstractions.

Thank you for your kind comments. Regards.

Bob Shour

    Dear Mr. Shour,

    Abstract "nature" does not have an "approach". Mathematics is incomprehensible to me. Reality is understandable to me.

    Joe Fisher

    Dear Mr. Shour,

    I have been trying to follow your essay, and have some problems understanding the calculation you undertake on p. 3. Your write:

    "Suppose that the average individual problem solving rate in a society of n individuals is x problems per time unit, and the whole society's problem solving rate is X problems per time unit. Is there a function F such that X = F(n)x?"

    On the assumption that every problem solved is solved by someone, then F(n) is just n: the total number of solved problems is the number solved (on average) per individual (which you call x) times the number of individuals. Perhaps you mean that there is a function x = F(n) which describes how the efficiency of problem solving by individuals is determined by the number of people in the society? Then your equation should be X = F(n)n, not X = F(n)x.

    I am also confused about the meaning of μ. You say μ is the number of other nodes (people) that a given person communicates with in a given time (on average). If that's right, then μ is just a number that characterizes the communication web of the society, and must be determined empirically as data. But you go one to worry about somehow solving for μ. Solving on the basis of what?

    It is also not clear why you make the assumption of isotropy. In a practical context, it would mean that each individual in a society communicates randomly: there are no groups that communicate strongly in-group and weakly out-group. That is not at all a realistic assumption. It does imply that one can rely on results from random graph theory, but random graphs are not reasonable models of actual human communications in a society.

    Maybe you can clarify a bit what you have in mind here.

    Regards,

    Tim Maudlin

    Dear Tim Maudlin

    Thank you for engaging with the essay. I will try to address your points.

    Forget about F for now. Start with X. X is a rate, not a number. For example, in 1657 there were 200,000 English words, and in 1989 there were 616,500. Assume criteria are the same for both counts. Then average English lexical growth is 3.39% per decade. The whole of English speaking society is involved in appraising (voting) on which words get used and which get discarded before making it into a dictionary. We have one possible collective rate of improvement in ideas. Can we corroborate the rate?

    The labor cost of lighting from 1750 BCE to 1992 improved at an average 3.41% per decade. All of networked society collectively decides which lighting improvements are favored and lay the foundation for the next technological improvement.

    Third, average IQs, though over a time period that is shorter, improve at about the same rate.

    Conclude: ideas that reflect widespread collective problem solving (per lighting, lexicon, average IQs) get better at the same average rate of about X = 3.4% per decade. The conclusion depends on interlinking and mutually corroborating concepts. This is a skeleton outline.

    Apply economic analysis. Just as the invisible hand of networked market interactions result in a market price for goods and services, assume the invisible hand results in an energy price for collective problem solving so that the information pay off for energy input required by collective problems is comparable. Then mathematical ideas used by society should be improving also at about 3.4% per decade.

    If today there were 100 English words, given society's collective problem solving rate of 3.4% per decade, we would expect there to be 103.4 words ten years hence.

    That is the end of what be called part One. If one accepts the existence of a collective problem solving rate, one has then laid the foundation for a much harder problem, is there an F such that X = F(n) x. In effect, this asks: how much smarter is society (what is F) than an individual? It turns out that one has to find F for each of two networks, one of brains and one of ideas.

    That leaves outstanding the issue of F, which is the number of degrees of freedom RELATIVE to the network's mean path length. Suppose for example the mean path length of physics profs is 3 and there are 9 professors. Then F for 9 people is 2, because one step of information transmission can (has the capacity to, not does) reach 3 people and one more step (each of the 3 can pass information on to a colleague) reaches 9 (if we don't duplicate the information paths). Why isotropy? That is a natural consequence of working with the mean path length. Since the average is the same for everybody (is that tautological?) then every average node has the capacity to transmit to the same number of average nodes per information iteration. By working with averages, isotropy is necessarily along (so to speak) for the ride.

    The mean path length is the average distance between members of society in STEPS. If you know the head of your department H and I don't but I know you, then H and I are 2 steps apart. How many mean path lengths does it take to span a system from one end to the other? The answer is log_\mu(n) where n is the population and \mu is the mean path length. Yes, but the mean path length \mu is also the average distance from one end to the other.

    This implies that the same energy it takes to go one mean path length confers a CAPACITY to traverse log_\mu(n) mean path lengths. One energy unit cannot both go one mean path length and at the same time several. So log_\mu(n) must be measuring the degrees of freedom of population n relative to the mean path length \mu.

    So F is a measure of capacity. Capacity is linearly related to F. If a brain has a larger network of concepts (eg 100 ways to prove the Pythagorean theorem compared to 1), then that brain's ability to solve geometry problems is greater.

    In effect F is the entropy of a network.

    My essay compresses many years work. I attempted to simplify the essay by way of a succinct sketch. I hope this reply helps. What have I left out in this explanation?

    Regards.

    Bob Shour

    Dear Tim Maudlin,

    This is a supplement to my initial reply.

    You mentioned strong in group connections etc. The calculation of F is of an emergent meta quality of a network. The clustering coefficient C is the average connectedness of nodes, which averages the effect of strong and weak in groups. So the C log (n) formula characterizes the entire network by utilizing averages. By using averages an enormous amount of information relating to individuals is compressed using only 2 parameters, C and the mean path length \mu.

    Consider the distribution or use of a finite amount of energy per time period (a rate) for 10 million households (of electricity), of 100 billions neurons in a human brain or 350 million brains (both energy via food). Use electricity distribution as a paradigm. We can calculate electrical use per capita per time period for 10 million households. Consider that 10 million degrees of use freedom per household. Now suppose all 10 million households each have a maximum rate of energy use during a year. The electrical distribution system does not have the capacity to deliver electricity to those 10 million households if they are all simultaneously at peak usage. But peak usage per household varies in time of occurrence. So the system only need to have capacity to meet the greatest average use. If a networked electrical distribution system has 10,000 supply nodes supplying 10 million households, the flexibility of capacity arises from the 10,000 degrees of distribution freedom.

    Similarly, for a brain's neurons, problem solving output capacity is affected by the degrees of freedom in the neuronal (or synaptic) paths available for problem solving. If there are 10 ways to get from a house to the office, the traveler can vary the route to minimize energy use. 10 ways to solve a problem gives more capacity to the problem solver than one way. To measure that capacity a log function relative to a mean path length gives a network capacity (as opposed for example, to a simple per capita energy use rate).

    Humans collectively aim for the most efficient solution.

    Regards,

    Bob Shour

    11 days later

    Hello. The question of collective intelligence is an interesting one, though I'm not sure it has much to do with the official topic of this contest (it would have better fit with last year's contest).

    However I am skeptical about the possibility to modelize it and establish any explicit formula about it. All what I consider to be clear is that increasing the population also increases the rate of technological progress but slower, and at a rate that is relatively lower as the population is big (i.e the advantage of a more numerous population vanishes when the population is already big enough so that being bigger no more changes much).

    About your formula, I disagree with your way to derive it. First if we believe your assumptions, that any individual has a rate of solving problems that is independent of the rest of the population, that all problems solved by individuals are actually different problems (not the repetition of each other), and that the solutions are going to spread in the way you describe, then what we get is that the number of solved problems is proportional to n (population) that are produced at a time and that take the time log(n) to spread. Then we would seem to get a flow of problems that would be not log(n) as you wrote but rather n/log(n) (dividing the number n of problems solved in parallel by the time log(n) it takes to spread) but that is not even true because log(n) is not a factor of slowing down of the flow, but of delay after which the solutions would reach the whole population. During this delay, the flow of solutions goes on. In a large time interval (much larger than spreading time), if solutions add up and do not repeat each other and are all going to be spread then the slowing factor is irrelevant, so that the factor is n instead of n/log(n).

    But I would also question your assumptions. It is common in science to have co-discoverers of important results, so that adding more researchers in parallel does not help. I discussed some disadvantages of overpopulation, including for speed of progress, in this text.

    Also, I consider that your model of spreading is not realistic. Seriously, do you know any example of solutions to problems that are spread in this way ? Human attention cannot be extended endlessly, to simultaneously verify many solutions found by others. The only process I see roughly behaving in the way you describe, is the process of Darwinian evolution, with the results of beneficial genetic mutations spreading across a population.

    Instead, if things happen ideally in an ideal world, any solution once found and verified by a few people comes to be published and thus instantaneously spread across the world, bypassing peer-to-peer communication.

    But we are not in an ideal world, so that in many cases, solutions once found are not spread at all and remain unknown. Especially because intelligence is not something common, and intelligent people are not always well-connected to each other, so that even after someone finds a solution, and even once it is explained to a few people, the information might not go further, as the mediatic space is occupied instead by a lot of rubbish which looks much more interesting in the eyes of the less intelligent people. I reported here my experience about this.

    Also, there are cases where a solution becomes known by the relevant community, but we have a persistence of a large separate community that remains ignorant about the solution. For example, the problem how to understand fundamental physics for so many purposes, has been solved by the community of physicists in the first half of 20th century, with general relativity and quantum physics, but still we have a persistence of a large community of cranks, including the majority of authors of essays in this very contest, who failed to understand these solutions because they are not good at maths, but to not feel ashamed of their failure to understand maths they need to mistake things as if the known mathematical solutions were wrong for no real reason but the fact these solution are mathematical so that it makes them obscure in their eyes.

    Similarly, in the 19th century and until the 1930's there were big monetary instabilities. Economists found ways to stabilize the money by central banks. However the Bitcoin community still never heard about that problem and solution, and keeps blindly believing that the instability of the value of the bitcoin is just due to the fact it is not popular enough and it will naturally become stable by the magic of being popular, just because, as usual currencies are now relatively stable since a few decades, the very existence of the problem remains completely ignored by people who specialize in cryptography and have no clue about finance.

    It is also possible that a larger population is a direct obstacle to innovation, as the larger number of people obliges a standardization of work where any deviation from the standard is becoming impossible. This particularly happens with the teaching system.

    On the other hand there are cases where a large population induces a collective form of problem solving where solutions work without being understood by individual members (or at least not by any majority of members). This especially happens with the "invisible hand" of free market that provides a sort of global optimization that does not need to be understood by any individual in order to work.

    Dear Sylvain Poirier,

    On your first paragraph. As a body of knowledge, math concepts and methods (which still grow) took much more energy than any one person could spend in a lifetime, and hence represents a disembodied higher intelligence which problem solvers use when they solve problems. Hence, math is effective because in a sense it is smarter than individuals.

    On your second. A population being bigger has slower change; a log function is not inconsistent with your point.

    On your third, the formula uses statistics about the entire network; the independence of individuals is not a required assumption. Co-discovery (your fourth) and redundancy are irrelevant; the rate of increase is of the entire network. Problem flow is analyzed by considering the networking of people and ideas, which have been measured.

    On your fifth, I suggested language and lighting. Ideas collectively made observably improve. What is a metric for improvement?

    On your sixth. A broadcast does not depend on peer to peer. The paper considers network effects, not just broadcasts.

    On your seventh and eighth: I respect the contributed essays so far because:

    (1) One can learn about (network with) other people's ideas such those about Bell's theorem, Kolmogorov complexity; the calculus of quaternions, and that is a valuable thing in itself.

    (2) Even though one may not understand or agree with all the essays or even any of an entire essay, some other people might benefit or get an idea, or in future might helpfully use some of those ideas. The forum allows ideas to meet, so to speak.

    (3) Even if none of the essays are entirely correct, and even if some are in parts entirely wrong, essay remarks, questions, and hypotheses may lead someone to a good idea, even if it is an opposite hypothesis, for example.

    (4) Estimation of someone's idea by a singe person, and even by a substantial majority, may be and often is incorrect. What seems wrong today, may be right in the future. That is the value of diversity of opinion, and the opportunity to network with different people and ideas.

    Regards,

    Bob Shour

    Your question on principles of accuracy, perfection, ambiguity and changing biological aspects of tales on whole is in your subject.

    Great concept driven!

    Sincerely,

    Miss. Sujatha Jagannathan

    Dear Sir,

    Your concept of development of language and mathematics are questionable. While evolution of information is well established, there is no proof of evolution of intelligence.

    Without defining intelligence, you cannot even remotely equate ants with human beings. At any moment, our sense organs are bombarded by a multitude of stimuli. But at any instant only one of them is given a clear channel to go up to the thalamus and then to the cerebral cortex, so that like photographic frames, we perceive one discrete frame at every instant, but due to the high speed of their reception, mix it up - so that it appears as continuous. Unlike the sensory agencies that are subject specific (eyes can only receive electromagnetic radiation, ears only sound, etc.); the transport system within the body functions for all types of sensory impulses. The same carrier transports the external stimuli from sensory agencies to the cerebral cortex and back as a command. This carrier is the mind. The existence of mind is inferred from the knowledge or lack of it about external stimuli. Only if the mind transports different external impulses to the brain for mixing and comparison with the stored data, we (Self) know about that (for the first time impulse received about something, there is no definite 'knowledge'). It requires an agent to mix these signals and convert them to electro-chemical information and submit to a conscious agent (operator) to cognize and utilize them. In perception, this task is done by a transitory neural activity in brain called intellect. Though, it is not directly perceptible, it is inferred from its actions - firing of positrons in specific areas of brain during perception. Each individual can develop his intelligence by learning from others, but there is nothing like collective intelligence - like a group performing a physical task, which is linearly additive.

    The people who constructed pyramids were not primitive. Before 3500 BCE, in India each letter of the alphabet was pronounced in at least 18 different ways and the meaning of the word depended on the specific pronouncement. Hence it was not written. There are highly codified formulas for this, which are still available. One book written by Panini is referred to by computer professionals even today for developing programming. All modern Indian grammars follow those methods in a much diluted format. Those people developed the number system including zero and also had mathematical treatises including highly developed geometry (called Shulva Sootra). At around 4th Century BCE, Chanakya in India compiled the earlier works on Statecraft and Economics, which is treated as authoritative even today. Please do not denigrate our ancestors.

    Regards,

    basudeba

      Dear Basudeba Mishra

      You mention 'there is nothing like collective intelligence'. 1700 years ago Pappus wrote about bees making hexagonal cells and attributed collective intelligence to them. A journal called Swarm Intelligence is devoted to that topic. Many books have been written about it. Some of them are mentioned in the reference section of the Wikipedia article on collective intelligence. The idea is that networking can sometimes lead to emergent effects, including intelligence. Ants and people both can network. Several of the inferences you read into the essay (pyramids, etc.) are neither in the essay nor intended. On the contrary, I agree there is much reason to respect legacy knowledge.

      Regards,

      Bob Shour

      Dear Sir,

      Collective effort (networking) is not the same as collective intelligence. Efforts are physical. Intelligence is conscious. We must distinguish between the two.

      Incidentally, nothing we write is original, but borrowed from our ancient knowledge. You can read our essay to get more details.

      Regards,

      basudeba

      10 days later

      Hello Bob,

      An interesting, clearly and well written contribution. I do not have much to quarrel with in your observation and conclusion, except that it seems to me that:

      1. There is Negative and Positive intelligence. Positive intelligence being the aspect that solves problems and Negative intelligence being the aspect that creates problems.

      2. The Total "swarm intelligence" can increase in time but has a zero sum value. (That is total positive IQ total negative IQ = Zero IQ).

      The implication is that as Positive intelligence increases (based on the number of networked neurons), Negative intelligence must also increase in tandem. In a bee colony, a bee selflessly carries food to the hive to feed the Queen bee, but a more positively intelligent but hungry human would rather eat that food than go hungry for the sake of the colony. Self interest replaces Group interest as Swarm intelligence increases. Anyway, that is my humble opinion.

      All the best in the competition.

      Akinbo

        Dear Akinbo Ojo

        Thank you for your comments.

        You have an interesting take on intelligence; maybe you're right. My essay's position is that intelligence can be thought of as the rate of problem solving. With IQ as a rate, there is no subjective element, i.e. positive or negative intelligence. But I agree that some human behavior in the world gives an impression of negative intelligence.

        Even given your realistic observation, I think that since our medicine, science, math and technology get better (we add newly solved problems), the net is not zero; we improve slowly. That is what I was trying to measure: the rate of improvement in ideas. One implication of the essay is that the average collective problem solving rate (which is a logarithmic equation) is 60 or 70 times that of an individual. Which makes me suspect that is why individuals have to work so hard to figure things out, and why overcoming what you call negative intelligence is sometimes such hard work: it is hard for individuals (and whole societies) to figure things out.

        Incidentally, I read your essay a couple of weeks ago, liked it, and appreciated that it was truly about foundational issues.

        Best wishes,

        Bob Shour

        9 days later

        I like your premise Bob..

        I make a similar point in my own humble essay, because I agree that Math knows things we have not learned yet. I also think that your point on the evolving efficiency of Math as a language is well taken, though I think the proof offered is a bit less than compelling. While it is true that our collective predictive capacity exceeds that of any one human, I am less than thrilled with the level of cooperation and collaboration I have seen.

        In a lecture I attended by Gerard 't Hooft, he suggested that some important discoveries and advances may never come, unless we can achieve an order of magnitude more integration between people of different disciplines. He said that not only should we have Physics people of differing specialties talking problems through, but the discussion should also include Math folks, Computer programmers, Engineers, Technicians, and others.

        In a lecture by Sau Lan Wu, she spoke of actually seeing that kind of cooperation at CERN, during the process that led to the discovery of the Higgs boson - where a large team of inter-disciplinary participants all contributed to the success of their efforts. But this is unfortunately an exception to the rule. While Math is a language that all these people could utilize, Bob; that makes them exceptional people, as well. I therefore see the applicability of Math and its usage among people as two separate issues.

        All the Best,

        Jonathan

          9 days later

          Dear Johnathan Dickau,

          Thank you for reading and commenting on my essay.

          On your points about collaboration: my essay looks at the result of uncoordinated emergent collaboration. For example, in language, there is no central committee mandating that 'cheers' should become a salutation in written correspondence such as emails. It has just become popular; likely networking via emails has a lot to do with that.

          Market prices also emerge.

          In mathematics, there is no committee directing research and planning how new ideas with help figure out problems in physics. So deliberate collaboration and emergent improvements in ideas have different mechanisms; both involve networking, but collaboration involves deliberate decisions to network and how to network. These ideas suggest the point you make, well taken: if ideas get better in the back ground, so to speak, what happens if we up our game and improve collaboration? I think we are improving our collective emergent problem solving capacity, mainly through the internet, and including wikipedia, arxiv, archive.org, and FQXi, for examples. Several essays in this contest (including yours, which I enjoyed reading by the way) have references to internet based resources. The internet increases the capacity of people to participate in problem solving. That seems to me give us a kernel of optimism.

          Regards,

          Bob Shour

          8 days later

          Bob,

          Thanks for sharing your ideas on the effectiveness of math and physics and pointing out the universality of math as a language and its evolution through people of various languages.

          I too deal with the connections of math, mind, and physics in the macro and the micro worlds to contribute to huge strides in DNA and simulating the BB with the LHC, and also a new field of quantum biology.

          Best regards.

          Jim

            Dear James Lee Hoover,

            Thank you for your comment.

            I read your essay, which gave a nice overview of connections of math and science, with some great quotes. I also noted your G_t formula on page 3 of your essay, which is an idea similar to the idea of the rate of problem solving discussed in my essay.

            Thank you, best wishes,

            Bob Shour

            Dear Bob Shour,

            I thought that your engrossing essay was exceptionally well written and I do hope that it fares well in the competition.

            I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

            All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

            Joe Fisher