Dear Mr. Wes Raykowski，

Can we express all the things in mathematics with elements (language, words, linguistic symbols… ) in common linguistics? Do both mathematics and common linguistics follow the same logic?

Yours sincerely,

 Geng

Hi Geng,

Re: “Do both mathematics and common linguistics follow the same logic?”

Yes and no. In my view languages and mathematics have common biological ancestor – they both originate from the way humans and their physical environment interact. In other words, they are different expressions of the same biological structure, as a result any differences between them are superficial and more related to various ways logic is used.

Re: “Can we express all the things in mathematics with elements (language, linguistic symbols… ) in common linguistics?”

If you are asking whether linguistics could explain mathematics, my answer would be negative. Based on my assumption that both languages and mathematics are different expressions of human biological reality, one should look for answers in biology (understood broadly). If in fact both common language and mathematics inherited the same logical structure this should be evident in both fields of human expression. However, finding those common structures is not easy as they are well hidden by the peculiarities of both types of expression and their historical development.

If I can use the issue of infinity so close to your heart, by studying mathematical expressions one can find out that the concept of infinity consists of two aspects, one involving never ending processes, and the other encapsulation which is a fancy word for thinking (and talking) about something as a completed process even if it is endless. This might seem to you illogical, but this is what the study of mathematical expressions suggests. The interesting aspect is that this kind of thinking is not only peculiar to infinities, but could also be found in other areas of mathematics, and in common language. 

Wes   

Hi Wes, I just came back from a business trip, sorry.

I once thought that in mathematics people use numbers, diagrams, sets,… to describe what we cognize to the things around us; while in common linguistics people use words, linguistic symbols,…  to describe what we cognize to the things around us. So they both do the same things by different forms------as you put it “they are different expressions of the same biological structure”. The problem is how we develop the “mathematical forms” or “linguistic forms”. What is your frank view point?

According to my studies, “mathematical forms” or “linguistic forms”, each can not describe clearly what we cognize to the infinite things around us-------that is why the potential/actual infinite debates will never be ended in present classical infinite related science theory system frame.

Yours sincerely, 

 Geng

Hi Geng,

Re: “According to my studies, “mathematical forms” or “linguistic forms”, each cannot describe clearly what we cognize to the infinite things around us”

To see what I meant in my previous message, you have to adopt the sensory perspective which assumes that - what we perceive as reality and its objects are the effects those objects have on our bodies. We call those effects sensations of vision, touch, etc.

There are two important observations one can make using this perspective. First, we do not experience objects (e.g. a chair, car, another person, or infinity), but the effects those objects have on our bodies, and ultimately our sensory system. Second, nobody can experience your sensations, the same way as you cannot experience the sensations of others. I hope you agree with both conclusions of the perspective I adopted.  

To express own sensations to others each individual has to produce material objects and place them in the (public) space where the objects could be accessed and engaged with physically. For example, if I feel like 'sitting' I could express this by constructing an actual chair capable of producing sensations I wanted to express. I can also point with my finger to an already existing chair. Alternatively, I can utter or type the word “chair” which ‘points’ to some prototypical chair. There are many problems such expressions. For example, no two people are likely to sit in the constructed chair the same way, and the chairs we point to may not be able to produce the sensations we have in mind. Written and spoken symbols are even more ambiguous, and so on.

The position I described above is similar to the theory called “radical constructivism” which assumes that

“the cognizing activity is instrumental and neither does nor can concern anything but the experiential world of the knower. This experiential world is constituted and structured by the knower’s own ways and means of perceiving and conceiving, and in this elementary sense it is always and irrevocably subjective.” (L P Steffe et al, 1996, Aspects of Radical Constructivism and its Educational Recommendations).

I tend to go one step further by defining our “experiential world” entirely in terms of sensations and the knower as a multicellular collection. But this is another story.

Please note the differences in our assumptions. The way you talk about mathematics, and infinity in particular, reminds the realist perspective which assumes that objects exist independently of human existence. If that is the case then all humans should perceive the object such as infinity more or less in the same way. This sentiment is evident in your arguments about infinity and infinitesimals – you want us to see the one and only true notion of infinity the way you envisaged it.  Your conclusion is logical but valid only when considered from a realist position.

RE: “that is why the potential/actual infinite debates will never be ended...”

The potential/actual infinite debates are not likely to end for various reasons, not necessarily because it is difficult to express ideas. There are other sources associated with human biology and sensations. We all know what it is like to have the body which has a boundary. On the other hand, we also know what it feels like to make repetitive acts which potentially can never end. Those two distinct experiences inform our understanding of infinity. You can see this in the expressions from everyday language and mathematics. Although possible, emphasising only one of those two experiences does not reflect human thinking.  

Re: “The problem is how we develop the “mathematical forms” or “linguistic forms”.”

I think that current mathematical definition of infinity is complete because it combines both aspects of human experience: bounded-ness of the human body; and potentially unending nature of repetitive actions. The same experiential structure could be observed in language, and even in fine art.  

Does my argument make any sense to you?

Wes

Hi Wes, thank you for your ideas. Now we are to the bottom of “science”------what is human science?

You are right that in my mind human science belongs to whole human (not only to a single person or part of human) basing on our biological structure (bodies, and ultimately our sensory system) and all the objects existing around us------our experiences (a kind of our cognizing fruits) are the effects those objects have on our bodies and ultimately our sensory system; but science is only a small portion (those belong to whole human) of all the cognizing fruits (knowledge------religion, experiences, culture, mathematics, infinite,…). So I really believe there is one and only one true (scientific) notion of infinity in human science.

Why I am so sure that current mathematical definition of infinity is far from complete because many “potential/actual infinite” related paradoxes are impossible to be solved in present classical infinite related science theory system frame. No one in the world now can give a self-justification answer to following question:

“Can we really produce infinite items each bigger than 1/2, or 100, or 1000000, or 10000000000,… from Harmonic Series and change Harmonic Series into an infinite series with items each bigger than any positive constants (such as 10000000000)”?

Yours sincerely, 

 Geng

Merry Christmas and Happy New Year!

Dear Wes,

Yes, the natural human languages are also some level of formalisation. However, in an attempt to resolve the deepest issue you run into the problem of undecidability primacy (in fact, the problem of causality). A formal system, which is a mathematics, is not only a consequence of the thought processes, and at the same time the reason. The reason of our universe, the reason for the origin of life, etc.

It is fundamental uncertainty. Therefore It is still need to intercede Platonism:

1. Ideas are the product of physical activity of the human brain so that they are also a particular configuration of matter in the universe.

2. Some things exist independently of the specific embodiments (the golden section number "e", for example).

It is believed that in the fundamental part of mathematics, the philosophical and linguistic approaches are unavoidable for analyzing and cognizing mathematics.

The question is, what remains from the object of study in the process of construction of hierarchy of abstraction until the complete abstraction from the sense. Cognition is arranged in such way that at any moment everything we come into contact with can be divided into understandable and transcendental constituents. This dichotomy is the first. At one extreme we have ready abstract relations, at the other extreme we observe concrete. On the physical level this all is the topological relationships in the information literally up to the structure of matter.

The definition of “science” decides that the balanced bridges between “abstraction and concrete”, ”subjective and objective”, “human and universe”, “theoretical and practical”,… are necessary and  important thought it is very difficult to build them. Those un--balanced bridges will lead us to unscientific cognitive fruits-------the errors and paradoxes in present infinite related classical science are typical cases.

Geng, yes, it is dificult, but another possibility does not exist, no matter how big problem.

In the twentieth century it was a period when with light hand of the structuralists everything literally tried reduce to the so-called "binary oppositions". Ultimately, in the Judeo-Christian culture it is associated with very old opposition between Good and Evil, that one without the other can not exist, they are mutually permeable and often are interchanged.

Rhetorical question, where is the concept of entropy?

https://en.wikipedia.org/wiki/Structuralism

Hi Vasiliy and Geng,

For the last twenty or so years I have been searching for those aspects of human constitution and behaviour which all humans have in common with each other, with animals and the inanimate matter. This is why I prefer the biological and physicalist view points over those of anthropology, sociology and religion. I think we have been paying too much attention to differences and too little to similarities. I do not deny the importance of differences and differentiation, but I am not obsessed with it the way postmodernists and poststructuralist are.

To find out what thought processes are likely to be common to all humans, I took an atomism - like approach. I assumed that ‘thinking’ can occur only in multicellular animals because only they could represent the external world internally by means of specialised cells capable of reacting selectively to some aspects of the material world outside living collections. This allows the collection to duplicate spatial relations existing outside and alter them by changing spatial relation inside the collection in ways that protect and promote its own integrity.

The really interesting issue is the problem of representing spatial relations. The answer has been known since 1950 when the first somatotopic maps where identified. My interpretation of the subsequent research is that spatial relations are duplicated by altering activation of individual neurons of the map. To cut the long story short, perhaps the most basic notions in human thought are those of intensity and its extent, both of which could be found paired up in many (if not all) human expressions including language and math.

Wes,

You are absolutely right saying "we have been paying too much attention to differences and too little to similarities." But when you immerse yourself in the understanding of this issue, you will find amazing and at the same time very simple things. Differentiation (process of dichotomy) plays a key role in the uniformity, you are talking about. It is deeper than the concept of atomism and affects the fundamental problems of cosmology and anything else.

The process of cognition is the mode of existence of the evolving structures. Only at the level of a man it is perceived as some special process. The evolving system is strongly nonlinear. Because of this and some other characteristics of our existence as part of it the uniform phenomena on different scales seem different behavior.

It is difficult to describe in a compact form, although the statement itself elegantly. The Mind is a fractal. In order to catch the general physics of the processes you will have to dive into such things as pink noise (long memory), self-organized criticality, the fitness landscape and multiverse cosmology.

Hi Vasiliy and Wes, happy to enjoy very insightful ideas and healthy way of scientific discussion.

@ Vasiliy,

I think without the effective “balanced bridges”, the "binary oppositions" approach is sure to bring endless debates and paradoxes to us------ too much on this side or that side, even not knowing what is each side (the typical example is potential infinite and actual infinite). I also agree with you that “The process of cognition is the mode of existence of the evolving structures” and I call this “evolving structure” metabolism as living organism has.

@ Wes,

It is true that “similarities” plays important roles in our research work because “similarities” are the sources of “abstractions (things with the same “nature” to human knower and the phenomena is abstracted as “equaltivity”)” in human science as well as many marvelous crosswise ideas in our scientific researches. Scientific studies on “equaltivity” is a newly opened up field in science and an ancient Chinese book YI JING (The Book of Changes) has presented this importance incisively and vividly.

Best regards,

Geng

Hi Vasiliy and Geng

Re: “The evolving system is strongly nonlinear.”

One of the things I was wondering about for a while is how humans understand such statements as “non-linear” change. The answer could be found, where else, in mathematics:

If         f(x) = x^2           then     f’(x) = 2x

If         f(x) = 2x           then     f’(x) = 2

If         f(x) = 2             then     f’(x) = 0

It looks that we understand exponential relationship (x^2) in terms of the linear one (2x) which is a fast ascending proportional line, which in turn is understood as a constant value line (2). In other words we tend to think of exponential or, more generally, nonlinear processes in terms of linear changes which in turn are understood as states, derivatives of which are zero. To summarise it, humans simplify complexity by conceptualising it in terms of proportional and constant experiences they can reliably simulate. This was one of the issues I tackled in my PhD dissertation.

Compare this line of thinking to a very complicated graph called “Hypothesis Mind Map (version1) you offer on your site. Your graph shows how rich the human thought could be. But if one carefully analyses each of those node concepts, one should be able to “reduce” all of them, so I think, to a small range of basic cognitive ideas.

Wes, I agree.

But I note that everything can be simplified to a certain limit. Since holistic system can not be decomposed into individual simpler components. However physical process which is an imprint in our entire formal mathematics really quite simple.

My associative map is just the tip of the iceberg for a group of people which is well aware and involved in the process. It is something like sitemap to find patterns. Most of the searches and reasoning focused on the hyperlink of the title of the diagram (unfortunately mixed in russian and english).

At first glance it is difficult to grasp how wide problematic issues it touches: the philosophy (of mathematics), Gödel theorems, the concept of infinity (static eternity), religions, the anthropic principle (proponents of string theory and quantum gravity broke spears of which), the uncertainty principle, the brain formation, the problem of the existence of lefties and dexters, introverts and extraverts, etc. (entropy in theory of information)

Hi Vasiliy, 

What are those patterns? Could  you described them in layman's terms? 

Best, Wes

I had in mind the general regularities, causal relationships between various phenomena and objects, as well as search for analogies between them.

Analogy is the key point in the process of cognition. They allow us to do an extrapolation from known object in the unknown. For example, Newton's analogy between the apple and the Moon allowed to extrapolate gravity on large scale and formulate the law of universal gravitation, etc. Analogies in conjunction with the principle of Copernicus still work not just in cosmology. Despite the fact that we are constantly expanding outlook. In a certain sense it is the reverse to dichotomy process.

Dear Professor Wes Raykowski and ProfessorVasiliy Fedorovich Komarov,  how do you think of following story (forgetting about “bigger” or “uncountable”):

Just take a tiny portion of rational numbers (elements) from Rational Number Set (1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …) and they map very well to all the numbers (elements) in Natural Number Set (1, 2, 3, 4, 5, 6, …, n …); so a lot of rational numbers (elements) from Rational Number Set are left in the “one—to—one element mapping between Rational Number Set and Natural Number Set”.

 So, Rational Number Set has far more elements than Natural Number Set.

Best regards,

Geng

Hi Geng, 

First, I am not a professor. Second, to say that "Rational Number Set has far more elements than Natural Number Set" you must bound both sets. And as soon as you do this you are dealing with numbers and not with infinities. 

Best, Wes

Dear Geng,

I am also not a professor, I am researcher.

A simple change of variable 1/rn = m results suggested sequence of rational numbers to the natural. This is equivalent set. They can not help in comparsion for long-known reasons.

For rational numbers is impossible to abandon the concept of the continuum. We have already discussed this in the thread that you created earlier. The problem of continuum hypothesis Aleph0 < ? < 2^Aleph0 there is not unreasonable problem.

In light of some recent thoughts I can add some visual and simple physical examples:

When integrating we dealing with one-dimensional segments with the length property but the result is a continuum of area.

You can also consider the square of the Peano curves. In the limit it also takes on a new quality - continuum of area.

It is not possible to directly compare qualitatively different objects.

Counterquestion. How can we break the integral object apart to get a set of, that can be compared with the other set of countable objects? We are dealing with the integrity - holistic property of continuum. Rational numbers live within.

With regards, Vasiliy.

Dear Mr. Wes Raykowski and Mr. Vasiliy Fedorovich Komarov, many RG friends are right that I am now really working in a “strange prime intuition way not believing some long-known reasons” because of so many members of suspended infinite related paradox families.

I had a very sad and trouble beginning in the first period of my study on the “very undoubtedly strict mathematical divergence proof of Harmonic Series”. As time being, I found the newly discovered Harmonic Series Paradox is in fact a member of 2500 years old Zeon’s Paradoxes family and more troubles came because I found myself in an unbelievable huge working field--------those defects disclosed by suspended Zeon’s Paradoxes family are there, they should be solved by us own.

My conclusion is: the “2500 years old huge black cloud of infinite related paradoxes over mathematics sky” is produced by too less intuition and ontology but too much formalism in present classical infinite related science system-----an integrating work is badly needed. So I have following 3 working principles:

1, pick up “prime intuition” in my studies on those defects disclosed by suspended Zeon’s Paradoxes family in present classical infinite related science system.

2, basing on the new achievements from “prime intuition of human science”, not to be “polluted” by some “terms and results” relating to those discovered defects disclosed by suspended Zeon’s Paradoxes family in present classical infinite related science system.

3, the new achievements from “prime intuition of human science” are right there unavoidable ( such as a revolution in “infinite”, “infinite related number system”, “infinite related limit theory”); they will be further developed someday by some people from different fields.

With regards,

Geng