The Janus of Mathematics by Burton J Smith

Dear Burt Smith,

Maybe, you have no idea how to change mathematics to better conform to reality. This would explain why you did not yet respond.

I would nonetheless appreciate if you could say more about Janus, the Roman god of door, standing for January, for in and out, for left and right. See my recent reply to Doug Bundy at 833 as to admit that my request is not unfounded.

Regards,

Eckard

Aloha Burt Smith,

You have chance lol it is probably beautiful there.

You are welcome, sincerely

Steve

Aloha Jenniver L.

Quite so; few individuals wish to be average. However, most like to know where the average is so they can decide what, if anything, to do about it. Since the average comes with a consensual agreement as to its characteristics most individuals can find enough wiggle room to shift themselves from above to below or vice versa, depending on subject matter.

Outliers, or the "lunatic fringe" as its sometimes called, present a different situation as they are events or items that are not suppose to occur with any regularity; yet often do. They are typically viewed as an inconvenience at best and disruptive at worst. But the fact remains it is

the "lunatic fringe" that is responsible for many, if not most, of the major changes, good or ill, that occur.

Sincerely

Burt Smith

Aloha Don:

One can add up the number of items in a junk yard and compare the total to the total of items in another junk yard and determine which one has more items. Not a great deal of information is obtained but of some value, especially if one is in the junk yard cleaning up business.

Haven't looked at your suggested site, but will.

Thanks for your comments

Sincerely

Burt Smith

Burt,

Yes in math 1plus1 = 2 This corresponds to the addition of your Junk items. Your use of the term "item" indicates that you have turned the junk into numbers.

But in physical reality I have reason to believe that 1 lb. of junk plus 1 lb. of junk > 2 lb. of junk.

I believe there is a big divide between physics and math. I also think we are poking at the same thing. And thanks for your reply.

Don Limuti

Aloha Eckard:

Believe it or not - I do have other demands on my time.

A brief discussion of the two faces of Janus and why it was applied to mathematics was pruned, along with several other items, so as to remain within the character limits of this contest. I had hoped that the discussion of pragmatic mathematics as opposed to the theoretical would suffice. I'm often wrong about such things.

You are correct in saying that I did not offer an alternative solution to mathematics as we presently know it. It would delight me no end to say that I had a solution, and for a slight consideration I would be willing to share it - but I don't.

I came to distrust mathematics in the 70's. The more I've worked with natural and ecotone systems (agriculture) the more I became convinced there was something basically wrong with math when applied to the natural world. As I mentioned in my essay, division in the natural world is very messy and is akin to death for any complex system; hence, it is not a proper operand for understanding the natural world. Multiplication as duplicator in addition is feasible, as is addition and subtraction. However, the information obtained is limited as to which pile is larger/smaller than another.

In your essay you have rejected analog computation as being out of date and too noisy. My experiences with analog computation in the 70's was that it was superior for modeling systems than were the digital behemoths of that time period. The vast sums that have been spent enhancing digital calculations make comparisons today difficult at best. Yet, there are groups working with analog; e.g. the Silicon Brain group at Stanford.(Ref, #54 in my essay.)

Other possibilities have to do with the way life changes over time, a slow morphing of form. Rivers flow and carry along all sorts of things besides water molecules; then there's the various forces found throughout our corner of the universe (electromagnetic, gravitation, etc.) and of course time. An orchestra combines the sounds from a large number of instruments into a unique sound - the composer starts the process and the conductor adds the finishing touches. Electromagnetic waves are capable of transporting such information vast distances; what we know about the reaches of the universe are carried on such waves. Are these, more natural (dare I say analog) phenomena, being investigated as possible calculation mediums? I hope so. The mathematics we have gets close on occasion - but close only counts in horseshoes.

Sincerely

Burt

Aloha Don:

Sorry, but the use of the word item does not automatically imply numbers. Item has several definitions, one of which is "an object of attention, concern or interest" which is the meaning I intended, and I thought was clear from the context - apparently it wasn't.

Adding a measurement (noun) to a numeral, I believe reduces the numeral to an adjective, and the onus of accuracy then falls on the noun, pounds in this case, to guarantee that "a pound is a pound the world around." Since a pound is, by definition, a fraction of the standard kilogram kept in France, the ultimate onus is on that platinum-iridium chunk - which I understand has loss a little weight since it was first cast. Not that it will change anything as there's a greater difference in the weight or an item from mountain top to sea level than in the few micrograms the standard lost.

And yes, I think we are, more-or-less, poking at the same things.

I have read the items at your web site. You've obviously spent a great deal of thought and time in the formulation of the theory. You're to be congratulated.

Sincerely

Burt Smith

If one can't step into the same river twice, one can walk with the river - go with the flow. It may not be exactly the same river one steps into each time but it's certainly closer than if one proceeds cross-wise.

Burt,

Wise words above. Your approach is different than mine. Mine borrows from models in positing the point of analogue reality. Yours less so.

Jim Hoover

Your analogy of natural flows and walking with the river is interesting. Could you expand a bit on this and how using a mathematics that flows with nature could alter science as we know it? How would this be different than the results of today's science which has it's own so called flow, a flow of continuously asking the same or similar question to demonstrate repeatability and achieve consensus? Would the results remain similar, but our perception, understanding, and use of the results be what is profoundly different?

Dear Burt,

Thank you for responding honestly. I agree on that a mathematical reduction of the the whole world, the wetware of the brain, and of other highly complex systems is inappropriate. I also agree with your insight that there is effectively no equality and no absolute symmetry in nature. Nonetheless I consider mathematics necessary and worth improvement.

Regards,

Eckard

Aloha Anon: Thank you for your comments. You've asked three good questions.

Q. Your analogy of natural flows and walking with the river is interesting. Could you expand a bit on this and how using a mathematics that flows with nature could alter science as we know it?

A. It is my contention that the very foundation of modern mathematics is built upon axioms and postulates that have little to no relationship to the natural world and are, at best approximations and at worse, misleading. Mathematics is based on discrete numerals; there is nothing fuzzy about a 4 or 9; because integers are defined to be constructed from monads which are defined to be equivalent to one another and hence discrete. There is nothing fuzzy about a meter - all meters are defined to be equal to one another. However, nature doesn't define anything to be equal to one another; everything changes with time. You are not the same person today you were yesterday, and you won't be the same person tomorrow. We don't skip, jump or hop from one scene to the next like a movie frame; we morph or flow, a continual, analog process. This is not unique to living organisms, the physical world does the same, but at a slower pace; the ebb and flow of tides, climate, even rocks disappear grain by grain until a pebble is left, and then that too is gone. At what point does a rock become a pebble, the pebble a grain of sand, and the grain a molecule? The lines we draw around things in order to call them the same are artificial boundaries that have meaning only in the human world, not in the natural one.

A river was first used as a metaphor by Heraclitus, and later by Plato. As a flow on earth it's a nice example, all sort of things can be carried along, some stopping for a spell, but eventually all ending up at some far off objective. As far as stepping in the same river twice, well as far as most are concerned one can do just that. We see the river, we give it a name, Mississippi River - sometimes it's turbulent, wild, other times placid and calm - but it's still the Mississippi River. We ignore the fact that as we stand in the shallows of the Mississippi River the water, and all that its carrying, swirling about our feet, is constantly changing. Hopefully, the concept of flows will make us more aware of the things that are continuously changing rather than concentrating on the river, which changes little from day to day and hardly at all in memory.

Q. How would this be different than the results of today's science which has it's own so called flow, a flow of continuously asking the same or similar question to demonstrate repeatability and achieve consensus?

A. It often appears that science does the same thing when replicating a previous experiment so as to verify or refute a previous one; and therein lies the rub. If an experiment cannot be replicated then how can one verify its veracity? Experiments utilizing inanimate objects can, usually, be replicated to a high degree and statistics applied. First, because inanimate objects typically change slowly over time, thus their similarities do likewise. So in a laboratory, where the many variables that thrive outdoors can be, more-or-less, controlled, one can flip a coin many, many times before anything different than a head or tail appear. However, take the experiment outdoors, over a mud puddle or sand box and, assuming a sea gull or dog doesn't grab the coin before it hits the ground, a third choice becomes apparent; the coin will occasionally land on its edge and remain so. Even physical experiments are not devoid of location.

Life might be considered as the physical world on steroids. Many of the same elements are involved but in unique ways. But life, unlike inanimate objects, has all sorts of behaviors that a coin can only aspire to. So obtaining the same level of repeatability with systems involving living organisms as with inanimate objects is much more difficult if not impossible. Without getting into the numerous ways statistics can be and are misused, suffice to say that the sample size and uniformity are quite important as to its validity; as mentioned in my essay the use of a sub-sub population class (undergraduates) to define human behavior. Most studies of natural attributes, humans in particular, end up with bulk of the sample grouped around a norm, with lesser numbers trailing off to either side; a distribution in the shape of a bell curve. We concentrate on the norm and those closely surrounding it, and all but ignore the outliers. Humans are unique, but some are more unique than others; for example, about one in sixty thousand of us, have the heart and other asymmetric organs on the opposite side. What medication, or procedure, might work for the majority might be fatal or un-performing for the outliers, or vice versa. If the sample is too similar then the results are only going to apply to those who are similar to the sample, and maybe not even then. Because there are near endless ways to group a sample, studies continue that appear to be addressing the same thing but are, in effect, looking at how it affects a different group. Soils are notoriously variable, back in the late 1940's and early '50's fertilizer studies were quite common; apparently little has been settled as they're still being conducted.

Q. Would the results remain similar, but our perception, understanding, and use of the results be what is profoundly different?

A. That would depend upon the system under discussion. Laboratory studies, where many variables can be controlled, would likely remain the same. Field studies would be different. Good, knowledgeable managers can generally move a complex, dynamical system towards, but not necessarily reach, a well thought-out and reasonable objective through continuous monitoring and timely reaction to inevitable changes - something that present mathematics cannot do. What the manager cannot foretell are all the incidental details that will be occurring as the system moves through time. I suspect that utilizing an analog-mathematics will increase the chances of getting closer to the objectives and will likely fill in more of the details. Our present economic system is based on a linear mathematics, which when applied to the natural world, is essentially that 'take and run' philosophy that helped us get to the agricultural age; it also got us to the present situations so eloquently discussed in the daily media.

Dear Sir,

We fail to understand the rationale behind your groupings. While the definition of Categorization appears correct, the definition of Generalization given by you fails the test of exclusivity needed for differentiating between different groups. In fact it is nothing but Categorization by another name. You define Generalization as: "Drawing conclusions from the shared characterizes of the items within a category", which makes Generalization a subset of Categorization. This is against the literal interpretation and common usage of these terms.

The above two nomenclatures are related to perceptions of different characteristics that define the objects. Since these characteristics have not been described individually, we have to assume that you refer to the totality of the identifying characteristics. This makes your third category redundant. You define "subitization or numerousness" as "The ability to rapidly ascertain if one group of items is larger/smaller than another". The terms "larger/smaller" are used only in dimension related descriptions, such as spread (one dimension), area (two dimensions) and volume (three dimensions) or descriptions related to the numbers associated with those objects. Further, the terms "larger/smaller" imply comparison with other objects, which is nothing but measurement and whose results are always expressed in terms of scalar quantities, which are nothing but numbers. Numbers are the perceived property of objects by which we differentiate between similars. After we perceive an object, if there are no similar perceptions, we designate the number associated with that object as one. If there are other similar perceptions, depending upon their sequence, we give each a name. These are the number sequences. You also appear to admit it when you say: "children, even babies have this ability", because even babies perceive, though they cannot use number sequence effectively. But then you contradict yourself when you say: "Numerals are not required."

If "subitization or numerousness" is related to numbers as described above, then it becomes one of the identifying characteristic of the objects and thus a subset of the earlier categories making its separate classification redundant.

Tallying is nothing but comparison, which is nothing but measurement and whose results are always expressed in terms of scalar quantities, which are nothing but numbers. Yet, you say "tallying does not require numerals, a mark or notation suffices" without realizing that it is a self contradictory statement. Numerals are nothing but notations. In fact you go on to support this view while arriving at an opposite conclusion.

When you say: "While subitization can help explain counting up to 4 and possibly 5, something else is required to count beyond that", you have unknowingly referred to a very important mechanism. After the perception of one, the perceptions of two and three arise very rapidly. Hence these are so named in Sanskrit. For this reason, even many animals show this prowess of counting up to three. The perception of four requires higher intelligence. But still it is not very difficult. The perception of 5 onwards requires definitely higher intelligence not only about numbers, but interactions. Use of fingers is nothing but a type of notation. Hence the reference to parietal lobe is without much significance. What you appear to be hinting at is the difference in the mechanism of perception of numbers up to 3 and may be 4 and the other numbers. You call the first category subitization. This classification reminds us of an anecdote related to Einstein:

Einstein had two dogs: one big and one small. He wanted to have a box for them to protect them from cold. He called a carpenter and told him to prepare the box with two doors for entry of the two dogs. The carpenter said that there is no necessity for two doors and one will suffice. Einstein said that two are needed because the big dog cannot enter through the small door. The carpenter replied that the small dog could enter through the big door. At this Einstein is reported to have said: Since the carpenter solved a problem which a Nobel Laureate like him could not solve, he deserved a Nobel Prize. It appears you are talking like Einstein and we are talking like the Carpenter.

Kindly review you article based on the above observation. The descriptions of historical developments by you are o.k.

Regards,

basudeba

Dear Burton,

Wisdom is more important than imagination is more important than knowledge for all the we know is just an imagination chosen wisely.

Please read Theory of everything at your convenience posted by me in this contest.

Who am I? I am virtual reality, I is absolute truth.

Love,

Sridattadev.

Aloha:

I have to admit that this is the first time anyone accused me of talking like the famed Dr. Einstein, nor have I ever been chastised by a plurality referring to themselves as talking like the famed carpenter.

Apparently you are not familiar with scientific papers and their use of references. Those small superscripts found scattered throughout the essay can be deciphered on the final page titled references. There you will find all the information needed to locate the referred paper or book . The phase, sentence or sometimes the paragraph preceding the superscript are, if not in the actual words of the referenced author(s), are his or her thoughts or ideas on the subject matter at hand. The original essay contained over 85 references. Unfortunately, when the paper was submitted it was rejected due to excessive pages of references. After consultation with the referee it was decided that I could reduce the font size and place all I could on one page, prune the rest; or withdraw the paper. So I reduced the font size to 8, pruned nearly 40 references from the reference list, and got them all on one page. However, I left the essay untouched, other than re-numbering the references, as it was a bit late to re-write it, even if that had been allowed.

About half of the pruned references were from the first two pages of the essay. I realized that the concept of mathematics as a human invention going back to Ardipithecus ramidus was likely to upset any number of individuals that for one reason or another held certain books or beliefs inerrant. Cultural paradigm are quite effective filters; so if you have a problem with my definitions I refer you to references 10,11, 12, 19 as well as numerous papers by Robert Sieger, Stanislas Dehaene, Andreas Nieder, George Lakoff, and Rafele E. Núñez - and Merriam-Webster's Dictionary as your definitions seem a bit esoteric.

You also appear poorly informed concerning subitization and tallying. I suggest you consult ref. #10, What counts by Brian Butterworth or "The Number Sense" by Stanislas Dehaene; or "The History of Mathematics," by David E. Smith; vol. 1. (this one's old but so is tallying).

I appreciated your comments, tho I do not agree with them. Many of the thoughts expressed in the paper are not exclusively mine, as I hope I've made clear. Some come from the frontiers of cognitive science, others from anthropology and animal behavior that are increasing not in agreement with the image of "the divine human." I have attempted to show that not only are there two faces of mathematics (pragmatic and theoretical) but also two faces of the world we live in; the real one and the abstract one we've built to protect us from our perceptions of the real one. The continual use of the tool that got us into our current predicament is not the solution.

Sincerely

Burt Smith

Aloha Sridattadev Kancharia:

I have read your essay but have no comment other than a belief is sacred to the believer, but not necessarily to others.

Thank you for your comments

Sincerely

Burt Smith

Dear Sir,

We thank you profusely for describing us properly. In fact we are "not familiar with scientific papers and their use of references". We have admitted so in the Author Biography in our essay.

We don't understand the cult of incomprehensibility perpetuated by the theoretical physicists of "mainstream science", which has failed to deliver anything new over the last several decades though the experimental physicists have done wonderful innovations. One example, LHC experiment designed by experimental physicists that proves Standard Model and SUSY developed by theoretical physicists wrong. We also don't understand the blind acceptance of "references" without applying one's mind to judge its veracity. In our unscientific terminology, we call it superstition. Add the cult of reductionism to the mix and you get a recipe to "scientific failures" and a sure way to lead to cozy life at public expenses by fooling the gullible public with incomprehensible "dogma".

We are not impressed by name dropping or references. We have given specific comments on your essay only because it is being debated on a public forum. We may not be a scientists or we may not understand much, but we have a right to comment. We expect you to give a specific reply to our comments (either proving it or disproving it) or else admit that you cannot reply to the points raised by us. The judges panel will judge you from your answers.

Regards,

basudeba.

It was nice to learn the term subitization. You use many names and people of history looking back (one face of Janus) but not much looking forward, (the other face).

As far as the loci of thought is concerned there are the ideas about the heart, and how the brain was not the center of thought. It was the heart, Aristotle as well as the ancient Egyptians seemed to believe. I found that such a striking idea when I first heard of it a few years ago, and is this what triggers the procreation and survival.....

Dear Burt,

I absolutely agree with you and respect your understanding, I just wanted to share this belief or realization that we are all connected. I wish you all the best as well.

Love,

Sridattadev.

Aloha Bert

A very different and refreshing new angle. and perhaps some of the truest words here in;

"The problems arising from a mathematics that cannot duplicate the natural world would be subtle and possibly not contentious, if only we were aware of its faults and planned accordingly."

Worth top marks in themselves. I've followed that through with logic for the model in my essay. It is an astonishingly falsifiable and vastly simplified physics, of SR from QM, yet shrouded from initial view by over reliance on maths. As a Mensa member I think you will see it, but it takes dynamic visualization skills 3 from 4 don't possess.

The solution is consistent with top essay Edwins, Regazas maths and many other reality based essays.

I hope you'll try, and comment.

Many thanks

Peter