Dear @Dejenie, let me recall BERTRAND RUSSELL,"Study of Mathematics:Mathematics, rightly viewed, possesses not only truth, but supreme beautya beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry." And this is very true! 

Thanks for bringing nice work of Eugene Wigner. Based on his work, the following article appeared: THE MIRACLE OF APPLIED MATHEMATICS by Mark Colyvan. 

The Unreasonable Effectiveness of Mathematics belongs here!

Dear @Dejenie, I do stand behind you repeating the words that we do deserve it. I could not imagine my life and life behind without mathematics.

http://colyvan.com/papers/miracle.pdf

http://schneider.ncifcrf.gov/Hamming.unreasonable.html

http://kwelos.tripod.com/unreasonableeffectiveness.htm

Mathematics are not magic. Mathematics is simply an extension of our everyday logic. Most of what we find in science is everyday and self-evident. It is true that, some times, mathematics can force our attention to solutions more rapidly and clearly than without such succinct symbolic language; but, in and in itself, it is no more magical than being conscious, nor does it explain anything about our world above and beyond what we know by simply looking around and contemplating. A proof: most of the ideas in modern science were known and thought of back in Hellenistic times and earlier still. 

Dear @Dejenie, let me recall BERTRAND RUSSELL,"Study of Mathematics:Mathematics, rightly viewed, possesses not only truth, but supreme beautya beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry." And this is very true! 

Thanks for bringing nice work of Eugene Wigner. Based on his work, the following article appeared: THE MIRACLE OF APPLIED MATHEMATICS by Mark Colyvan. 

The Unreasonable Effectiveness of Mathematics belongs here!

Dear @Dejenie, I do stand behind you repeating the words that we do deserve it. I could not imagine my life and life behind without mathematics.

http://colyvan.com/papers/miracle.pdf

http://schneider.ncifcrf.gov/Hamming.unreasonable.html

http://kwelos.tripod.com/unreasonableeffectiveness.htm

https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

Mathematics is a planning tool for human achievements. In the context of engineering, it is very important for all growth and implementations, to see the changed world with respect to last century. Maths is so fundamental to science right from the basics of our living to all upgrades in medical instrumentation, telecommunications, space technology etc.,

Hi, Dejenie

Thank you for the good topic bringing us to a field of fundamental work in mathematics and science. I totally understand and agree that Eugene Wigner asks the question “What is mathematics?”, It is right to the point. in fact we are often at a lost: “What is science?” --------- “What is mathematics?”.

So, dear Mr. Dejenie A. Lakew , I think the deep structure idea of your question is “What is science?” --------- “What is mathematics?”, and your question might be put in the way of “Is the effectiveness of science a miracle, we humans do not understand and do not deserve?” 

Yours, Geng

It is pity that we human have been not doing enough on the philosophy of mathematics comparing with the philosophy of physics-----more basic concepts are confusing in mathematics than those in physics and we even have thousand years’ suspended paradoxes in mathematics. We were told when we were young: “just do, it doesn’t matter whether you understand or not”, and we now tell our students exactly the same. I believe that this is also a reason why we sometimes wonder “Is the effectiveness of mathematics a miracle, we humans do not understand and do not deserve?”

Dear Ljubomir,

Thank you for the  links that are pertinent to the topic. Indeed Bertrand Russell was one of the staunch lover and promoter of mathematics. The quote you mentioned is one testimony of his love of  mathematics  describing  it as ."... without appeal to any weaker part of our nature ... "  a very sharp description of the rigidity and uncompromising nature of mathematics. 

As you have said, indeed we mathematicians know it how mathematics works and physicists, engineers, applied mathematicians as well and  we deserve to utilize it. 

Mathematics touches every aspect of our life. It is not scarce. How can it be a miracle?

Mathematics is not magic, on the contrary is a self-consistent theoretical building, which, provided that we have found a good approximation of natural reality, can describe that reality in a self consistent way.

Mathematics have led us to a pleiad of wrong inferences about the natural reality of our local universe, although we cannot reject the possibility that their claims could be present at another local universe.

This is the main difficulty: to be able to distinguish between mathematical beauty in general and our given local universe's reality.

.

Dear  Dejenie A. Lakew, 

Effectiveness in mathematics is very useful, is used often and they do not look like a miracle.

http://www.academia.edu/6888389/Applicability_Indispensability_and_Underdetermination_Puzzling_Over_Wigner_s_Unreasonable_Effectiveness_of_Mathematics_

Regards, Shafagat

Dear Dejenie A. Lakew,

A mathematician has its view of mathematics. The realists, like Goedel, see truth as conformance with reality. The mathematical realism has sets as an expression of conformance with reality. Thus in mathematics we have already an expression of effectiivenness of mathematics.

Best Regards, César

Dear All,

Thank you for your keen and valuable answers. 

Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects The evolution of mathematics can be seen as an ever-increasing series of abstractions. In general, the evolution trend of advanced mathematics (high level abstraction) was governed by seeking analytical models and solutions for different phenomena, due to lack of  computing capability of now-a-day electronic computers. With the existence of computers much of these phenomena can be modeled numerically with low level abstraction (no need for higher mathematics).

Dear Dejenie and Dear RG fellows,

Very optimistic vision for the mathematicians!

Here is a statement of Charles Dodgson famous mathematician and author, under the name of Lewis Caroll, of Alice in wonderland. So also a successful writer!

Kind regards

Marg

Effectiveness of mathematics is not a miracle, actually it is part and parcel of everyday life.

Dear Ting,

A faultless life is the one that is not been lived. A life utterly of faults is the one that does not aspire to get to the good and truth of things.  

Dear Demetris,

Indeed mathematics clears and gives way to find the truth of things than to simply believe something that is not. Well said!

Dear Barbara,

Very true, mathematics enables us not only to understanding things around us but to lead our lives better. No action of our lives that is devoid of what we made it to be optimal decisions and decisions in most cases are made based on our logical capabilities - which is the machinery of mathematical thinking. 

Dear Cesar,

Well said, the effectiveness of mathematics is self evident and built within and reflected from without.. 

Given the topic and the mathematics of quantum physics, I'm going to opt for a superposition state.

On the one hand, it's somewhat problematic to wonder about the success of mathematics in describing reality when

1) After almost 3,000 years of mathematical development, most of the time we can only approximate real-world phenomena via mathematical models and all to often our models and even mathematics (as developed thus far) fail.

2) Asserting that mathematics is so incredibly effective at doing what we spent a few thousand years developing it to do is like saying that language is amazingly effective at communicating concepts about the natural world: that's what it does (only language emerged organically while mathematics was custom built to effective).

3) The failure of the bulk of Hilbert's program, realized especially in the groundbreaking studies by Turing and Gödel, Rosen's relational biology and various proofs and so forth from those influenced by him (particularly by AH Louie) suggest that living systems aren't computable and therefore aren't particularly suited to exact mathematical models.

On the other hand, there is nothing other than the weak anthropic principle to explain just how much mathematics can explain. Despite poor nomenclature, "chaos theory" has shown repeatedly the effectiveness of mathematics because such systems actually exhibit a particular kind of order.

Out of the many countable and uncountable infinite number sets, we find that a few constants appear everywhere in models of natural phenomena.

We developed mathematics in part so that we could effectively describe system dynamics. But why should this have worked as well as it has? Even "laws" we know are wrong (F=ma) are still amazingly effective in almost all applications.

Finally, over many years and in particular the past decade or so, physicists have increasingly relied on information theory to describe all reality. For many, this is not just a convenience but reflective of the incredible way in which information theory encapsulates all there is so far as the sciences can determine.

At times I have tried to figure out my position here, but inevitably I find myself swayed by arguments from both sides, not unlike the way I have continuously failed to determine what I believe the "correct" interpretation of quantum mechanics is. After all, in QM and modern physics more generally a central problem is that our mathematical description are not, as they are in classical physics, readily relatable to the states and properties of the physical systems they seek to describe.

Dear Dejenie,

I work with  Mathematicians and Physicists I think, that Maths and Physics can structure thinking, these students are able to solve the  nonstandard tasks not only  in Maths, but also in life. I'm astonished by their mental speed. There is a connection between Maths and intellectual abilities.They have developed logical, prognostic, deductive, abstract, critical, analytical thinking, and a good memory. They are highly motivated in learning Math, and have good results. I think a motivation (pleasure, interest) is the main driving force. On the other hand, we should develop other kinds of their thinking through arts, reading, foreign languages.They like performing.In my University, the faculty of Physics  was a leader in singing and playing the guitar.

How do you define the effectiveness in this case?

Dear Natalia,

I define the effectiveness of mathematics as Eugene Wigner and others defined it, its innate capacity in describing natural phenomena uniquely and foretell their future behaviors, but I  and  every scientist in this thread argued eloquently the reasons behind its effectiveness, " the reasonableness of the reasonable effectiveness of mathematics" than its "unreasonable" or "magic" contrary to the reality.  

Dear Irina,

You are absolutely correct in your assessments. Mathematics gives the power not only to do mathematical discourse and physics but also to properly address other challenges of life as well, since it is a discipline of critical thinking and reasoning. 

Mathematics also increases the general effectiveness of life!

In the context of education, I recognize the effectiveness of mathematics in being the basis. One needs to learn pure math before jumping into other subjects. In my younger ages math. seemed "magic"   to me. Bu it's "magic" even now, when I am able to connect via the solution of the radiation balance equation, the condensation of nitrogen in the lab. and its condensation on Pluto. About 50 AU away !!! And conclude that nitrogen there condenses as a transparent layer (as in the terrestrial lab), which can explain observations (see our article on Pluto on this Research Gate).

According to my experience, Mathematics is the most effective language of science. It facilitates the quantitative description of the objects and their behavior. It is an effective quantitative description tool. And so It is the complementary of the textual qualitative descriptions of the objects and their behavior.

Dear Dejene and Dear RG fellows,

I follow Albert Einstein that said:

"I think and think for months and years. Ninety-nine times, the conclusion is false. The hundredth time I am right" 

To get the good and the true a lot of mistakes is required.

In science sometimes you can reach good and valuable results, but in life this is more difficult unless you become selfreferential.

Nice regards

Ting Fa Marg

Why mathematics? What ever the subject it is, the required is how we raise questions!

I think it is the fixed and wonderful logics (relations) between or among mathematical things that make “mathematics a miracle”. We human have mathematical things as our cognitive fruits, discovered those “fixed logics (relations)” and built mathematics, of cause we do understand and do deserve.

 Cheers, Geng

Dear Tang and dear all,

Indeed unlocking scientific truth requires quite a number of empirical tests which in most cases results may not be convincing enough and we have to redo the process until we uncover the truth. However the mathematics we do in the process to confirm these  truths  are done consciously knowing it with an intent  to validate the truth we sought. The difficulty with life is that the variables or parameters involved in most cases are out of the control of the person and therefore to get optimum of life is almost impossible and therefore every person lives an approximate, a compromised life not true optimal life that is sought. 

The most painful thing about mathematics is how far away you are from being able to use it after you have learned it.

James Newman

Like theoretical philosophy and applied philosophy, we have theoretical mathematics and applied mathematics. They are there and serve us human in time.

One of many @Saeed's plagiarism (pure copy/paste). ResearchGate should be aware of such bad practice!!!

Here is the original:

"THERE IS A story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. "How can you know that?" was his query. "And what is this symbol here?" "Oh," said the statistician, "this is pi." "What is that?" "The ratio of the circumference of the circle to its diameter." "Well, now you are pushing your joke too far," said the classmate, "surely the population has nothing to do with the circumference of the circle."

Naturally, we are inclined to smile about the simplicity of the classmate's approach. Nevertheless, when I heard this story, I had to admit to an eerie feeling because, surely, the reaction of the classmate betrayed only plain common sense. I was even more confused when, not many days later, someone came to me and expressed his bewilderment [1 The remark to be quoted was made by F. Werner when he was a student in Princeton.] with the fact that we make a rather narrow selection when choosing the data on which we test our theories. "How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nevertheless, explains just as many phenomena as the present theory?" It has to be admitted that we have no definite evidence that there is no such theory..."

https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

Thank you Dear Ljubomir,

There was a similar case where a person was copy posting the actual posts that were on the thread site itslef and I asked the person what he meant, is it either endorcing or commenting on what it is said. If it is endorcing, there is a like button to promote it, if it is to repost it because of more liking, then that has to be quoated and indicated in a proper manner,  but the person did not reply. The same thing is with dear Saeed and he did not reply so far for my comments regarding such cases. It is not only here, it is in some of my thread questions that he did that. 

Dear Ljubomir Jacić and Dejenie A. Lakew,

Very interesting versions! RG members must avoid copy paste.

Yes, that is true. But, dear @Subhash, you have upvoted plagiarism on the previous page!!!

Dear Ljubomir,

It is very unlikely people here at RG checks whether the post is from someone(copied) or his/her own and that is why people simply read it and put a mark when they like it. It is good that you checked that out and indicated to me. 

It is very easy task dear @Dejenie. I do check when I see the answer with an excellent English from non-English speaking person! 

Dear Dejenie,dear Ljubomir,dear Geng,dear Subhash, dear All,

A very interesting work by dear Markovic G.Doko.

http://www.academia.edu/3759446/Some_Typical_Examples_of_the_Application_of_Didactic_Principle_of_Polyformism_in_the_Teaching_of_Mathematics

Dear Irina,  

Thank you for sharing the work of dear Markovic Doko on the theory of what he calls poly-form in teaching mathematics. He illustrates his arguments with several interesting example. 

Dear Arno,

Well said,  understanding the actual truth of things around through logic and reasoning  indeed creates ecstasy of the mind and thereby stability of thought processes. Therefore mathematicians enjoy that innate ecstasy and  stability from doing mathematics.  

Dear Mr. Dejenie A. Lakew,

It is good for us to be aware of the fact “we human sometimes do not understand and do not deserve the effectiveness of mathematics” and look for some reasons.

Discussing something is important but I think the most important thing is doing something: to do something in the philosophy of mathematics and the foundation of mathematics.

Mathematics is our human’s, we are able to build our mathematics understood and we can deserve the effectiveness of mathematics.

Yours sincerely,

Geng

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

http://ctpweb.lns.mit.edu/physics_today/phystoday/reasonably1_406.pdf

Dear Geng,

I am not sure if I understand you, are you affirming the idea that we humans do not understand mathematics and do not deserve to utilize its effectiveness or just explaining what the statement means? 

If it is on the affirmative, then I will say this. It is because we understand mathematics that we formulate it, develop it and utilize it in studying natural processes to the extent of being effective qualitatively. The true strength and power of mathematics as an effective  structure in studying natural processes is not a miracle or magic but from the logical consistency and the process of reasoning that are followed, starting from  the astute observation of the fundamental, natural  basic axioms up on which it is built. 

Philosophy of mathematics is a philosophical observation, dictation and argument made for the sake of scholastic satisfaction from being outside of mathematics, while mathematics is done solely by mathematicians and applied mathematicians of natural sciences.    

Dear Krishnan, 

I am a little hesitant to use the word miracle for mathematics, as the literal meaning of a miracle is something that does not obey laws of nature but happens suddenly, while mathematics is different from that description. You are correct and we all agree in appreciating the power of mathematics and the necessity of humans to always develop more new mathematics so that problems get better understanding and better solutions, which is always an ongoing endeavor, since what we understand through mathematics is an approximation to the actual truth and therefore solutions to problems are approximations with that degree of understanding.   

Dear Mr. Dejenie A. Lakew, thank you for your frank point of view.

I really mean the negative understanding in my post and the following is my frank ideas:

1, in the infinite related part of mathematics (a very big potion of mathematics) we human really do not well understand mathematics and do not deserve to utilize its effectiveness, newly discovered deep hidden Harmonic Series Paradox is a typical example.

2, philosophy of mathematics has a lot to do with the foundation of mathematics and the infinite related part is a typical example.

In my humble idea, philosophy of mathematics is not just a philosophical observation, dictation and argument made for the sake of scholastic satisfaction from being outside of mathematics------- the newly discovered Harmonic Series Paradox of “strict mathematical proven” Suspended Zeno’s Paradox has strongly proved that philosophy of mathematics is not just a kind of chat at one’s leisure.

Yours sincerely,

Geng

Dear Geng,

First, the concept of infinity is unambiguously defined and well established in mathematics and serves very well in a meaningful and effective way in almost every part of mathematics but specifically in analysis and its precursor calculus.

I have said in my previous comment on your thread question regarding your assertion of the Zeno paradox as a paradox in implementing the concept of infinity and convergence. Zeno's paradox is not actually a paradox in mathematics but of understanding motion of a physical object in time. Indeed a moving object with a certain speed covers  a certain distance (evolution in time), and the faster the object the longer the distance it covers in a fixed specified time length.

But Zeno did not know that and considered a moving creature as just a point on a number line and simply considered to go from one initial point to the next adjacent point, of course there are infinitely many points to trace as there are infinitely many rational numbers that are dense in any non degenerate interval, but that is not the case in motion. if you drive your car with 45 miles/hr uniformly, then you will cover 45 miles in one hour, you do not need infinitely many hours or you do not need to travel infinitely many halves of 45 miles. 

The zeno's paradox  is not also a paradox per se in its own, since the faster  rabbit will over run the tortoise after some time, unless the tortoise is already placed at the end line which in that case is not a match.   

Dear Mr. Dejenie A. Lakew，

The biggest trouble in present infinite related mathematics is the very much ambiguously definition of “infinite”. Some people insist we can have only one definition of “infinite” in science but others argue that we can have many definitions of “infinite” with different natures in science (at least two: “potential infinite” and “actual infinite”): “Infinite”, “potential infinite”, “actual infinite”, “potential infinitesimal”, “actual infinitesimal”, “potential infinite-big”, “actual infinite-big”, “Infinite related numbers”,…: INFINITE R is more infinite than INFINITE N--------infinite real number set is more endless, more limitless and more infinite than infinite natural number set because it has more elements than infinite natural number set.

In front of “infinite something”, one at lest have two troubles:

1, “potential infinite” or “actual infinite”

2, how much “endless, limitless and infinite” it would be.

(a),When paying equal attention on “potential infinite” and “actual infinite”, we meet the uncompromised logical contradiction and the typical case is the “strict mathematical proven” ancient Zeno’s Paradoxes and modern Paradox of Harmonic Series.

(b), When paying more attention on “potential infinite” and neglect “actual infinite”, we meet all the ideas and theories of “all infinite things are the same endless without any inborn and numerical natures” and the typical cases are all the ideas and theories of the countable infinite sets.

(c), When paying more attention on “actual infinite” and neglect “potential infinite”, we meet all the ideas and theories of “all infinite things are not the same endless, they have inborn and numerical natures” and the typical cases are all the ideas and theories of uncountable infinite sets: more infinite, more more infinite, more more more infinite,…; higher infinite, higher higher infinite, higher higher higher infinite,…; bigger infinite, bigger bigger infinite, bigger bigger bigger infinite,…; supper infinite, supper supper infinite, supper supper supper infinite,…; Continuum Hypothesis,…

Thank you very much.

Yours,

Geng

Dear Mr. Arno Gorgels，

Thank you very much for your kindness. 

Yours,

Geng

In present classical “multi--infinite--definition” related science, when facing “infinite something”, one at lest have two troubles:

1, “potential infinite” or “actual infinite” or “super infinite” or “sub infinite” or “secondary infinite” or “lower infinite” or…?

2, how much “endless, limitless and infinite” it would be?

So, just follow someone’s ideas doing something in present classical infinite related science:

1, we are wrong in doing something although we didn’t know why, just do (such as all the family members of “strict mathematical proven” Suspended Zeno’s Paradox);

2, we are right in doing something although we didn’t know why, just do (such as most of the successful work in analysis and set theory).

Something miracle？

Something deserve?

 Dear Mr. Arno Gorgels，

Thank you very much for your frank point of view and encouragement. 

But why the potency is limited?

 Yours,

Geng

 Dear Mr. Arno Gorgels，

I think science (mathematics) is our human’s, it should be understandable, explainable and without contraction.

 Yours,

Geng

Why math? JHU mathematician on teaching, theory, and the value of math in a modern world!

"I think there's a perception that some forms of math are problems just for the sport of it, like a crossword puzzle or Sudoku. They exist to challenge your brain, but they won't help cure cancer or put a man on Mars...

Without any context of why you're learning something, yes, it becomes just a bunch of problems that you need to solve. There are certain concepts, constructions, or equations that you're never really sure if they are going to show up anywhere or not. But presented in context, math has a number of real-world applications. But the problems themselves have value. Even at the primary school level, you can play Sudoku games and teach the logical structure of things. If you give people a sense that there's a creative nature to mathematics, then at some point, even at the primary school level, they can learn and wonder at some of the logical structures they're building. That is something they take with them the rest of their life, a curiosity about how things are structured logically..."

http://hub.jhu.edu/2015/11/17/richard-brown-why-math-qa

Dear Ljubomir,

Very true that there are parts of mathematics that is solely for the sake of  mathematics, to augment it and create new mathematics  until some particular time of applications of these used mathematics is reached. The expectation is that an abstract structure that is mathematically consistent and valid will somehow signify some natural process or behavior that we did not know before. 

Lobachevsky sums up this through his words  " No part of mathematics how ever abstract it may be that will not be put to the service of science and thereby mankind" . When we look at abstract mathematical structures and objects that are indispensable for natural sciences today, were initially treated as  absurd and odd to be considered even in mathematics let alone to reasonably describe a physical phenomena. 

It's the only miracle we can count on.