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RECENT POSTS IN THIS TOPIC

James Hoover: on 4/15/15 at 3:24am UTC, wrote Bob, I am revisiting your essay and am thankful you checked out mine. I...

Joe Fisher: on 3/31/15 at 15:16pm UTC, wrote Dear Bob Shour, I thought that your engrossing essay was exceptionally...

Bob Shour: on 3/30/15 at 15:13pm UTC, wrote Dear James Lee Hoover, Thank you for your comment. I read your essay,...

James Hoover: on 3/30/15 at 4:48am UTC, wrote Bob, Thanks for sharing your ideas on the effectiveness of math and...

Jonathan Dickau: on 3/22/15 at 3:10am UTC, wrote Thanks Bob.. Have Fun! Jonathan

Bob Shour: on 3/20/15 at 15:55pm UTC, wrote Dear Johnathan Dickau, Thank you for reading and commenting on my essay. ...

Jonathan Dickau: on 3/12/15 at 3:53am UTC, wrote I like your premise Bob.. I make a similar point in my own humble essay,...

Bob Shour: on 3/2/15 at 18:45pm UTC, wrote Dear Akinbo Ojo Thank you for your comments. You have an interesting...


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FQXi FORUM
October 15, 2019

CATEGORY: Trick or Truth Essay Contest (2015) [back]
TOPIC: Mathematics knows things about physics that we don't (as yet) by Bob Shour [refresh]
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Author Bob Shour wrote on Jan. 16, 2015 @ 21:50 GMT
Essay Abstract

Mathematical concepts compress solutions to problems involving natural and other phenomena. Hundreds of generations of human society have tested those solutions for their accuracy and utility and have continuously refined them. Those concepts likely often reflect fundamental aspects of the universe including phenomena not yet modeled, perhaps not yet identified. Mathematics therefore has predictive power. The accumulation of solved mathematical problems that we possess represents distilled intelligence that enormously exceeds that of any individual human being.

Author Bio

Since completing law school in 1976, I have continued to learn about mathematics and physics.

Download Essay PDF File

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Harlan Swyers wrote on Jan. 17, 2015 @ 15:32 GMT
Bob,

I like this idea that math has some predictive power as to what new physics will be. I am also keen on the idea of "problem solving capacity" you discuss in the paper. I was wondering if you had any thoughts on the idea of whether we gain benefit from "free-thinking" such that we allow individuals to choose what problems they want to solve, or should it be more directed? Some might argue that allowing someone to think of all the potential plot consequences of an episode of "Friends" or "Star Trek" is time wasted, but then one might counter that such thinking may have helped people in their approaches to problems they encounter in their own lives. Curious as to your thoughts.

Best,

Harlan

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Author Bob Shour replied on Jan. 18, 2015 @ 15:24 GMT
Dear Harlan Swyers,

Thank you for your comments. Free thinking has two aspects. Think about anything and see where it leads. A directed question invites a diversity of responses, one of which may add insights. Both seem important to me.

Regards,

Bob Shour

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Edwin Eugene Klingman wrote on Jan. 18, 2015 @ 02:56 GMT
Dear Bob Shour,

A most fascinating essay. Most essays are such that is not difficult to find significant areas of agreement, and specific details upon which we disagree. Yours is hard to disagree with, in general. From your key assumption that "brains networked over time have more problem-solving capacity than individual brains", combined with the idea that 'compression' over time is a powerful means of increasing efficiency, you discuss inference sets (with examples), Hayek (optimal allocation) and a fascinating discussion of language compression (with examples).

Then your examples of relative numbers of neurons, in ants, in humans, in ant societies, and in human societies, lays the framework for the hypothesis:

"Maybe networked neurons don't care where they are located when they exercise their degrees of freedom."

With this first-order approximation you proceed to analyze society's problem-solving rate. Again, fascinating!

You tie concepts of entropy from Shannon to Clausius, to analyze the function F(n) which relates the problem solving capacity of a society of n individuals to one individual.

You are to be congratulated on a most interesting and most enjoyable essay, one certain to do well in this contest.

My best regards,

Edwin Eugene Klingman

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Author Bob Shour replied on Jan. 18, 2015 @ 15:27 GMT
Dear Edwin Eugene Klingman,

Thank you for taking the time to read, comment on, and summarize my essay. Feedback is appreciated, and I am grateful when it is positive. I am glad you enjoyed the essay. I hope to have the chance in the future to improve the description and presentation of the ideas in the essay.

Best wishes,

Bob Shour

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John C Hodge wrote on Jan. 18, 2015 @ 15:47 GMT
You mention ants.

The context of this contest suggests an interesting observation about ants.

Humanity started building the perfect arch in the 1500s or so. This took considerably math ability, physics ability and many other human intellectual abilities.

Ants for their lack of neurons have been building perfect arches for millions of years. They substituted rules and evolution to do the job.

Their rules are (1) everybody starts to build a mound of mud. (2) after a while, groups of 3-5 identify the tallest mound and build on that while abandoning the others (3) after a while, they start adding mud to build toward the closest mounds. (4) when the mound meet they fill in the wall. As the tensions work their way, some mud falls but some stays to make the perfect arch.

Mankind should not get to enamored with their methods.

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Gary D. Simpson wrote on Jan. 24, 2015 @ 00:50 GMT
Bob,

A very interesting essay. Thank you for sharing these ideas.

I guess it is true ... two heads, or perhaps, 2.7 heads, are better than one.

I'm not sure if there is a word for it, but the fact that you have used a mathematical model to quantify why mathematical models are useful and effective is ... well, simply delicious.

The use of the number of words in a language as a surrogate for the complexity of a society is quite clever I think. There could be some second order interactions that cause error though. For example in the German language, it is common to make a new word by combining two other words ... as an example, consider "thought-experiment".

I have to wonder though, is it possible for technology to cause a step change in the effectiveness of a society at solving problems. For example, was there a step change when the printing press was invented or when the library was conceived? Possibly we will see such a change due to the computer or parallel processing? I will tell you, from my own experience, my ability to create, understand, and solve new problems is not limited by computational capacity. It is limited by my ability to have insights into the problem being considered.

Best Regards,

Gary Simpson

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Author Bob Shour replied on Jan. 25, 2015 @ 14:35 GMT
Dear Gary D. Simpson

Thank you for your comments.

On ‘step changes,’, the biologist Stephen Jay Gould similarly noted that evolution is also not smooth but has punctuated equilibriums. Whatever common thermodynamic principles underlie emergence might provide an answer to both points.

Suppose that there is a steady supply of energy to an ecosystem, a stable average use...

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Gary D. Simpson wrote on Jan. 25, 2015 @ 21:07 GMT
Bob,

The accuracy of your analysis of my approach to solving problems is amazingly accurate. If I were paranoid, I would think you were a fly on the wall in the room where I work. The main emphasis is upon legacy knowledge.

Any idea that produces accurate predictions must contain some element of truth. You may have opened the door for better usage of mathematics by the less technical disciplines.

Best Regards,

Gary Simpson

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Christophe Tournayre wrote on Feb. 1, 2015 @ 20:45 GMT
I enjoyed your essay. In my view, the extrapolation you are making is interesting because it opens questions.

One of them is : why do you stop calculating collective intelligence with life forms? For example, you introduce the collective intelligence of an ants colonie.

They are billions of computation happening on earth every seconds. It is not life forms, it is physical properties. Why cannot they be considered as intelligent computation too?

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Author Bob Shour wrote on Feb. 2, 2015 @ 01:07 GMT
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

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Author Bob Shour wrote on Feb. 2, 2015 @ 22:04 GMT
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

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Joe Fisher replied on Feb. 3, 2015 @ 16:18 GMT
Dear Mr. Shour,

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

Joe Fisher

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Member Tim Maudlin wrote on Feb. 3, 2015 @ 21:06 GMT
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

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Author Bob Shour wrote on Feb. 4, 2015 @ 01:36 GMT
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

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Author Bob Shour wrote on Feb. 4, 2015 @ 15:26 GMT
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

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Sylvain Poirier wrote on Feb. 15, 2015 @ 12:18 GMT
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...

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Author Bob Shour wrote on Feb. 15, 2015 @ 18:57 GMT
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

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Sujatha Jagannathan wrote on Feb. 16, 2015 @ 09:27 GMT
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

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basudeba mishra wrote on Feb. 18, 2015 @ 09:36 GMT
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...

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Author Bob Shour replied on Feb. 18, 2015 @ 14:51 GMT
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

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basudeba mishra replied on Feb. 20, 2015 @ 05:53 GMT
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

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Akinbo Ojo wrote on Mar. 2, 2015 @ 15:00 GMT
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

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Author Bob Shour replied on Mar. 2, 2015 @ 18:45 GMT
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

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Jonathan J. Dickau wrote on Mar. 12, 2015 @ 03:53 GMT
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

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Author Bob Shour replied on Mar. 20, 2015 @ 15:55 GMT
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

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Jonathan J. Dickau replied on Mar. 22, 2015 @ 03:10 GMT
Thanks Bob..

Have Fun!

Jonathan

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James Lee Hoover wrote on Mar. 30, 2015 @ 04:48 GMT
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

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Author Bob Shour replied on Mar. 30, 2015 @ 15:13 GMT
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

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Joe Fisher wrote on Mar. 31, 2015 @ 15:16 GMT
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

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James Lee Hoover wrote on Apr. 15, 2015 @ 03:24 GMT
Bob,

I am revisiting your essay and am thankful you checked out mine. I find that I did not rate yours, something I usually do to those I enjoy, so I am rectifying that. I like the concept of networked brains and math representing a composite of our knowledge in a distilled fashion.

Jim

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