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Author
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Topic: New Approach to Physis
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E. E. Escultura
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Member # 1872
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posted 03. February 2006 05:24
I just came across this interesting website and I want to contribute some ideas. I have always wondered why physicists spend most of their time as mathematicians, mainly, solving equations. In doing so they leave a lot of questions unresolved. For example, what is the basic constituent of matter? What is the structure of the electron? What is charge? What is gravity and what is its medium that allows two cosmological bodies at great distance apart to pull each other? Why do cosmological bodies spin?
Wouldn’t it be more fruitful to change the approach in physics to try to understand how nature works by finding the laws governing its motion? We really do not have to start from nothing. We can take off from the first law of thermodynamics and, perhaps, modify and improve it to pave the way for the discovery of other laws of nature.
Some recent observation by the Hubble can push us towards more discoveries. For example, it is known that matter steadily forms in the supposedly empty Cosmos at the rate of one star per minute. This rate has been amplified by the discovery of a nascent galaxy a few months ago. The jet outflow of hot gas spewed by the core of a nascent galaxy travels at speed 75 times that of light. The rate of expansion of our universe has been calculated from Hubble’s law to be of order of magnitude 10^20 km/sec and still accelerating at 10^(-10) km/secsec. Trying to resolve these questions and explain these natural pheneomena serve as catalyst for discovering other laws of nature. Then the task of the physicist is no longer to compute but to discover the laws of nature and explain natural phenomena.
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Christopher D. Beling
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Member # 723
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posted 03. February 2006 20:33
quote:
I have always wondered why physicists spend most of their time as mathematicians, mainly, solving equations.
The reason as I see it is
(i) Pragmatic - most physicists have to do practical things these days (with their known functional equations) in order to make a living. Sad I know.
(ii) Positivistic - since the time of the Vienna school of the philosophy of science - scientists have tended to believe less that they can attain an ultimate truth and believed more and more that they should only penetrate the natural world to the limit where they could predict all phenomenon - in other words they are fairly satisfied with what they have got! Maybe there is also some frustration at not being able to dig deeper easily.
quote:
Wouldn’t it be more fruitful to change the approach in physics to try to understand how nature works by finding the laws governing its motion? We really do not have to start from nothing. We can take off from the first law of thermodynamics and, perhaps, modify and improve it to pave the way for the discovery of other laws of nature.
Actually, not all physicists are engaged in practical - technological work. There are still those that think about more fundamental things such as the nature of particles, the foundations of quantum mechanics and the nature of space and time. Sadly the number in these areas is dwindling and ideally should be more.
The fundamental principle behind the 1st law of thermodynamics is quantum mechanics and the uniformity of time. This was shown by Emmy Noether
but surely this is not the end - for surely we must continue to ask why quantum mechanics takes the form it does!
The fundamental principle behind the 2nd law of thermodynamics seems to be the principle of causality - see the work of Denis Evans and Debra Searles in Advances in Physics . The second law deals with the increase of entropy (disorder) over time - i.e. that systems develop to states of maximumal probability - the microstate (at any instant) being of smallest probability.
The 3rd law of thermodynamics seems to be a fairly straight forward corollary to the 2nd law . It says that entropies tend to zero only at absolute zero temperature.
The 4th law of thermodynamics is also I believe an extension of the 2nd law, and as demonstrated by William Dembski [See for example No Free Lunch p 166] is based upon information theory. It states that in any system containing Complex Specified Information (CSI) that the CSI will either remain constant or decrease in time.
Hope this stimulates you and others to give their views on the fundamentals of physics. - Chris
[ 05. February 2006, 19:56: Message edited by: Christopher D. Beling ]
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Melvin H. Fox
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Member # 1684
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posted 07. February 2006 13:11
Professor Escultura,
I have read your paper “Countably infinite counterexamples to Fermat’s Last theorem.” The questions raised there and in your post here run deep into the way scientist think and operate. I can’t say that I agree with your conclusions in the paper but the maths is impressive.
As far as physics is concerned, I believe Chris makes two valid points with regard to attitude and agenda. It seems to me that we are all just a little to comfortable with Rn and its application to reality. Is this not your implication on page three of your paper where you write:
quote:
A fundamental defect of mathematics we are raising for the first time is lack of distinction between the subjective universe of thought and concepts, on the one hand, and the objective universe of their representation, on the other. What exists between them is a mapping and no logical conclusion can be drawn about someone else’s thought. Therefore, only the objective universe is accessible to collective study and axiomatization that can be communicated with precision among individuals and the proper subject matter of mathematics is the representation of the subjective in the objective universe. For instance, what does a theorem in number theory refer to: the concept of number or the numeral? Until this time, mathematicians did not bother with this question that has decisive influence on mathematics and science.
Please feel free to correct me if I have misinterpreted your meaning.
-Mel
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Edgar E. Escultura
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Member # 1878
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posted 08. February 2006 05:06
Dear Professor Fox,
Your interpretation is absolutely correct and I couldn’t agree more with you and Chris regarding the attitude and agenda of the physicist. I was guided by the same perspective until I had done some work in mathematics, especially, the charactertization of undecidable propositions. I felt that there was a better approach to physics than the present mathematical modeling. In the next post I’ll take the challenge by Chris to take on some fundamental issues.
E. E. Escultura
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Christopher D. Beling
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Member # 723
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posted 09. February 2006 19:56
Dear Professor Escultura,
It is nice to be in contact with someone at UP!. I looked up your Interesting paper mentioned by Melvin. It is quite long and complex (and I am not so good at maths). Would you mind me asking - if you could summarize your main points - it might help and stimulate us in our response to your position on the connection between maths and physics. Many thanks - Chris
[ 09. February 2006, 20:03: Message edited by: Christopher D. Beling ]
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Edgar E. Escultura
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Member # 1878
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posted 10. February 2006 06:54
Dear Professor Beling,
Not at all and I would be glad to give a summary of the paper. But let me first post what I committed to do in my earlier post.
There are physical concepts that are not directly verifiable in nature. Among them is latent energy. Only its observable impact is verifiable. One example is the latent energy of gasoline which gets converted to heat when ignited.
Moreover, nature does things with optimality. For instance, a tree develops nested fractal structure of its roots to maximize absorption of nutrients from the ground. The profile of a traveling wave on water surface is sinusoidal because that is the optimal balance between motion and symmetry. Perfect symmetry precludes motion and motion breaks symmetry. The circle has perfect symmetry including angular symmetry with respect to the center. However, when it is rotated uniformly, breaking its angular symmetry, and the position of a point in it is plotted against time the graph is sinusoidal which retains some symmetry. We shall consider other examples of optimality in nature. For now, we include optimality in the modified statement of the first law of thermodynamics that I call energy conservation law:
The sum of latent and kinetic (ordinary or visible) energy of a physical system is constant and in the interaction with its surroundings and other physical systems it maximizes energy gain and minimizes energy loss.
I consider this the most fundamental law of nature without which our universe would plunge into chaos. This implies that all other laws of nature must be consistent with it and if some natural phenomenon appears to contradict it another law must be found to reconcile it with energy conservation.
Chaos is mixture of order none of which is identifiable. An example is the initial rush of trillions of air molecules into a tropical depression. While each molecule is subject to the laws of nature it is impossible to identify, monitor and predict its motion. However, energy conservation will stabilize this system into a vortex, an ordered physical system called tropical cyclone.
I would appreciate critique of my formulation by colleagues.
I guess this is just long enough for a single post and so I'll post the summary next time.
Cordially,
Eddie Escultura
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Edgar E. Escultura
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Member # 1878
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posted 10. February 2006 12:32
Dear Professor Beling,
As I promised I shall highlight the important points in the paper noted by Melvin. In fact, I shall extend it to my mathematical work. The whole pursuit was catalyzed by Fermat’s last theorem. I wondered why this problem could not be resolved and concluded that inadequacy of present mathematics is to blame. I undertook a critique of the underlying fields of FLT, namely, foundations, number theory and analysis, particularly, its base space, the real number system, and I found the following:
1) Recognizing the ambiguity of thought and its concepts, as David Hilbert did almost a century ago, I agree with his proposal that mathematical concepts should be symbols bound together by consistent set of axioms. Recognizing that ambiguous or ill-defined concept is a source of contradiction and that contradiction invalidates a mathematical space, I move further and require that every concept be well-defined by its axioms. A concept is well-defined if its existence, properties or behavior and relationship with other concepts are specified by the axioms. “Existence” is stressed because vacuous concept, i.e., vacuous expression defining it, is ambiguous. Any proposition involving ambiguous concept is ambiguous. One implication here is that the decimals are distinct from binaries or triadics or base five numbers since they have different properties well defined by different sets of consistent axioms. There is additional requirement on decimals to avoid ambiguity: every digit must be known or computable, i.e., “determinable” uniquely.
2) Among the ambiguous concepts are infinity and large and small numbers (depending on context). Infinity in the sense of being unbounded or nonterminating (e.g., nonterminating decimal) is ambiguous. In the former ambiguity comes from the impossibility of enumerating or identifying all the elements of unbounded set. Therefore, any proposition involving the universal or existential quantifier may be verifiable on unbounded set. In the latter ambiguity comes from the limitation of computation.
3) Since distinct mathematical spaces are independent, each space being well-defined only by its own axioms, they are independent. Therefore, the rules of inference must be specific to and well-defined only by the axioms of the given mathematical space. It follows that any proposition involving concepts from distinct spaces is ambiguous. Consequently, proof relies solely on the axioms of the given mathematical space and unverifiable proposition is inadmissible as an axiom since
it does not insure the certainty of the conclusion of a theorem.
These requirements are the remedy for the inadequacy of foundations, especially, mathematical logic. The next post will deal with the weakness of the real number system and its extension to analysis.
Cordially,
Eddie
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Edgar E. Escultura
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Member # 1878
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posted 11. February 2006 09:39
I would like to make a crucial correction to my statement in the previous post,
"Therefore, any proposition involving the universal or existential quantifier may be verifiable on unbounded set."
The corrected sentence should read,
"...any proposition involving the universal or existential quantifier may NOT be verifiable on unbounded set."
Eddie Escultura
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Atom
Member
Member # 1840
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posted 11. February 2006 12:11
Ok, that line had lost me. I thought maybe you meant the quantifiers could be used to somehow remedy the situation.
Thank you for the summary. I agree with your main points on axiomatization and valid proof.
[ 11. February 2006, 12:11: Message edited by: Atom ]
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Edgar E. Escultura
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Member # 1878
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posted 11. February 2006 19:14
I need to make a correction to item 3) of my previous post that says,
"Since distinct mathematical spaces are independent, each space being well-defined only by its own axioms, they are independent."
It should be,
"Since distinct mathematical spaces are well-defined only by their respective consistent axioms, they are independent."
Let me now proceed to the real number system and how to make it free from ambiguity and contradiction. It is clear that its axioms (see for instance Royden's Real Analysis, MacMillan, 1973, pp. 31 - 32) do not meet the requirements we have set here for well-defined mathematical space. The idea is to avoid any ambiguity in the initial phase of the construction. Let me denote the new space I am constructing by R* with operations + and x which will be well-defined by the axioms.
Axiom 1. R* contains the basic integers 0, 1, ..., 9.
Initially, these elements are ill-defined since only their existence is assured. But the remaining two axioms will complete their well-definition. They will also well-define the rest of the elements of R*, namely, the terminating decimals of the form, N.a^1a^2...a^n, where N is an orinary integer that ranges through 0, 1, ... For the moment we don't know what a sequence is and so the definition is valid only for finite N (terminating decimal).
Axiom 2. The addition table.
Axiom 3. The multiplication table.
These tables specify the behavior of the terminating decimals and the relationship between them including comutativity, associativity, etc. We now well-define a nonterminating decimal as a standard Cauchy sequence N.a^1a^2...a^na^(n+1)...,
where every digit is known or computable, i.e., there is some algorithm or rule for determining it uniquely. There is now an element of uncertainty or ambiguity here because even if every digit, say, the kth digit, is computable not all of them can be stated or actually computed. The irrational pi is well-defined since every specific digit can be computed from its series expansion. Another example of well-defined nonterminating decimal is a normal number where every digit is chosen at random from the basic integers. An important number here is the dark number d* = 1 - 0.99... whose nonstandard Cauchy sequence is given by
d* = 0.1, 0.01,...
Like the infinitesmal of standard analysis thre is uncertainty in it being defined in terms of a nonterminating decimal, the difference is: d* is well-defined. Just like the infinitesimal, d* is smaller than any specified new real number but it would be a contradiction to say that it is the smallest real number because that does not exist. (By the way, all the well-defined real numbers are new real numbers).
The ambiguity here, however, is confined and does not led to contradiction because every Cauchy sequence is approximated by its nth term, which is well-defined, at margin of error 10^(-n). For instance, pi = 3.14159... is approximated by 3.141 with margin of error 10^(-3).
There are more details about the new real numbers on my thread, Contradiction-Free Mathematical Spaces, in Sci Math.
Cheers.
E. E. Escultura
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Edgar E. Escultura
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Member # 1878
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posted 11. February 2006 20:15
Sorry, I'm quite careful with my concepts but quite careless with my typing. The Cauchy sequene,
"N.a^1a^2...a^na^(n+1)...,"
should have been writtin,
"N.a_1a_2...a_na_(n+1)...,"
and the terminating decimal should have been written,
N.a_1a_2...a_n
E. E. Escultura
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Edgar E. Escultura
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posted 12. February 2006 16:55
Let me wrap up the summary of the major points of my mathematical work.
1) Since the known digits and the set of algorithms for finding the digits of the numbers are finite but unbounded so is R*. Therefore, R* is discrete and any function over it is discrete-valued. Therefore, R* and the space of functions over it is the appropriate mathematics for physics and computer science, especially, simulation, because nature is discrete and computing is digital.
2) The known new real numbers have natural ordering (lexicographic ordering) and are neither ambiguous nor contradictory.
3) The nonterminating decimals have contained ambiguity in the sense that they are “approximable” by their respective Cauchy terms to any desired margin of error.
4) R* is enriched by the new integers d* and N.99. .., where N = 0, 1, ..., are the ordinary integers. They are called new integers because the mapping 0 -> d*, N -> (N-1).99... is an isomorphic embedding of the ordinary integers into the new real numbers.
5) The isomorphism of the integers into the integral parts of the decimals resolves the fundamental flaw of number theory since the integers did not have valid axiomatization.
6) It is shown in my paper (The mathematics of the new physics, Appl. Math. Comp., 130(1), 2002, 145 - 169) that d* and 0.99... belong to the open interval (0,1) but d* and 0.99... cannot be separated from 0 and 1, respectively.
7) 0 behaves almost entirely like d* and 0.99... behaves almost entirely like 1. Thus, if x is neither 0 nor a new integer,
0 + x = x, 0x = 0, 0^N = 0;
d* + x = x + d* = x, d*x = d*,
(d*)^N = d*,
(0.99...)(0.99...)
= (1 - d*)(1 - d*)
= 1 - 2d* + (d*)^2
= 1 - d* = 0.99...;
by mathematical induction, we have,
(0.99...)^n = 0.99..., n = 1, 2, ...;
if x is not a terminating decimal,
x(0.99. . .) = x(1 - d*)
= x - xd* = x - d* = x.
8) The countable counterexamples to FLT, i.e., the solutions, for n > 2, of Fermat’s equation,
x^n + y^n = z^n,
are as follows:
x = (0.99...)10^T,
y = d*,
z = 10^T, T = 1, 2, ...,
and kx, ky, kx, k = 1, 2, . . .
I would be glad to explain any ambiguity in this summary.
Cheers.
Eddie
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Christopher D. Beling
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posted 12. February 2006 20:12
Hi Eddie and Melvin,
I wonder if you could give your opinions on this interesting paper of two colleagues of mine on the foundations of Quantum Mechanics which I believe is consistent with your views on the importance of numbers and discrete spaces.
quote:
Einstein's criticism of the Copenhagen interpretation of quantum mechanics is an important part of his legacy. Although most physicists consider Einstein's criticism technically unfounded, we show that the Copenhagen interpretation is actually incorrect, since Born's probability explanation of the wave function is incorrect due to a false assumption on "continuous probabilities" in modern probability theory. "Continuous probability" means a "probability measure" that can take every value in a subinterval of the unit interval (0, 1). We prove that such "continuous probabilities" are invalid. Since Bell's inequality also assumes "continuous probabilities", the result of the experimental test of Bell's inequality is not evidence supporting the Copenhagen interpretation. Although successful applications of quantum mechanics and explanation of quantum phenomena do not necessarily rely on the Copenhagen interpretation, the question asked by Einstein 70 years ago, i.e., whether a complete description of reality exists, still remains open.
[ 12. February 2006, 20:18: Message edited by: Christopher D. Beling ]
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Edgar E. Escultura
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posted 13. February 2006 11:12
Hi Chris and Melvin,
quote: Einstein's criticism of the Copenhagen interpretation of quantum mechanics is an important part of his legacy. Although most physicists consider Einstein's criticism technically unfounded, we show that the Copenhagen interpretation is actually incorrect, since Born's probability explanation of the wave function is incorrect due to a false assumption on "continuous probabilities" in modern probability theory. "Continuous probability" means a "probability measure" that can take every value in a subinterval of the unit interval (0, 1). We prove that such "continuous probabilities" are invalid. Since Bell's inequality also assumes "continuous probabilities", the result of the experimental test of Bell's inequality is not evidence supporting the Copenhagen interpretation. Although successful applications of quantum mechanics and explanation of quantum phenomena do not necessarily rely on the Copenhagen interpretation, the question asked by Einstein 70 years ago, i.e., whether a complete description of reality exists, still remains open.
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I am in full agreement with the quote's view that
continuous proability as description of reality is invalid mainly because reaality is discrete. With regard to Einstein's question whether a complete description of reality exists or is possible I would answer in the negative. Ultimately, description rests on computation which has its limits. But, if the question is whether nature can be explained fully in terms of its laws, I would say, yes. That is what Einstein had been searching for and now pursued by Hawding and others.
The fact that the applications of quantum mechanics have been remarkably successful, especially, in the area of technology generation reveals that we have reached a high degree of accuracy in in our description of reality. However, we have just scratched the surface of possibility. A new methodology, called dynamic modeling that explains nature in terms of its laws would yield a new generation of technology. This would require, using this methodology, extending the domain of quantum mechanics to what is called dark matter. Then this extended field is what I call quantum gravity.
Just like the steam engine that was invented before the development the science of thermodynamics, the magnetic train was invented before the development of quantum gravity which can fully explain how it works. There are other new technologies in the works based on quantum gravity.
Eddie
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Edgar E. Escultura
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posted 13. February 2006 11:13
Hi Chris and Melvin,
quote: Einstein's criticism of the Copenhagen interpretation of quantum mechanics is an important part of his legacy. Although most physicists consider Einstein's criticism technically unfounded, we show that the Copenhagen interpretation is actually incorrect, since Born's probability explanation of the wave function is incorrect due to a false assumption on "continuous probabilities" in modern probability theory. "Continuous probability" means a "probability measure" that can take every value in a subinterval of the unit interval (0, 1). We prove that such "continuous probabilities" are invalid. Since Bell's inequality also assumes "continuous probabilities", the result of the experimental test of Bell's inequality is not evidence supporting the Copenhagen interpretation. Although successful applications of quantum mechanics and explanation of quantum phenomena do not necessarily rely on the Copenhagen interpretation, the question asked by Einstein 70 years ago, i.e., whether a complete description of reality exists, still remains open.
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I am in full agreement with the quote's view that
continuous proability as description of reality is invalid mainly because reaality is discrete. With regard to Einstein's question whether a complete description of reality exists or is possible I would answer in the negative. Ultimately, description rests on computation which has its limits. But, if the question is whether nature can be explained fully in terms of its laws, I would say, yes. That is what Einstein had been searching for and now pursued by Hawding and others.
The fact that the applications of quantum mechanics have been remarkably successful, especially, in the area of technology generation reveals that we have reached a high degree of accuracy in in our description of reality. However, we have just scratched the surface of possibility. A new methodology, called dynamic modeling that explains nature in terms of its laws would yield a new generation of technology. This would require, using this methodology, extending the domain of quantum mechanics to what is called dark matter. Then this extended field is what I call quantum gravity.
Just like the steam engine that was invented before the development the science of thermodynamics, the magnetic train was invented before the development of quantum gravity which can fully explain how it works. There are other new technologies in the works based on quantum gravity.
Eddie
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