It is true that mathematics is, in a way, a branch of physics.
Our last question: "Is it true that mathematics is a product of physics?" has been denied by most mathematicians, while most physicists remain unanswered.
This question is more daring, and its answer is also:
How many mathematical problems can be solved within the physical framework, but never within the mathematical one?
Once again, we offer this extraordinary examples comparing the field of physics with that of mathematics as a present to our young generation of MSc and PhD students, to guide them for their future choice in physics and mathematics.
Unlike Google search and Wikipedia
[(Google search: No, mathematics is not a product of physics; rather, they are distinct, though closely related, disciplines, with mathematics serving as the fundamental language and tool for physics to describe, model, and predict natural phenomena. While physics explores the universe through observation and experimentation, mathematics independently studies abstract structures using logical reasoning. However, significant mathematical advances have often been inspired by physical problems, demonstrating a reciprocal relationship where physics provides problems for mathematicians to solve and mathematics provides the framework for quantifying and understanding the physical world.)]
We offer this extraordinary or groundbreaking analysis comparing the field of physics with that of mathematics, as a present to our young generation of MSc and PhD graduates.
How many examples show that certain mathematical rules and solutions exist and are possible in physics, but not in mathematics?
It's simple.
First Example
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What is the volume of a four-dimensional hypercube?
Mathematics suggests that the volume of a four-dimensional hypercube (tesseract) with side length "a" is 8a^4.
On the other hand, matrix string physics B (the statistical theory of Cairo techniques) predicts the volume of a four-dimensional hypercube (tesseract) with side length "a" to be simply a^4.
The question arises as to why the answer from physics is true while the answer from mathematics, if it exists, is false.
Ultimately, we would say:
Mathematics is, in a way, a branch of physics.
We continue to demonstrate how many mathematical solutions are possible in physics, but not in mathematics.
Second Example
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We demonstrate once again how many cases where mathematical solutions are possible in physics, but not in mathematics.
This time we are analyzing a problem that, surprisingly enough, has more to do with physics than with mathematics.
The problem we are all familiar with is solving a system of n non-uniform simultaneous linear algebraic equations:
a11. x1 + a12. x2 . . . . + a1n. xn = b1
. . . . . . . . .
an1. x1 + an2. x2 . . . . + ann. xn = bn . . . . . (1)
The beauty and elegance of this problem lies in finding a general solution that fits all non-infinite values of a and b.
The mathematical formula or solution to such a problem is impossible.
It simply does not exist.
In contrast, matrix chains B have the following matrix/tensor for the case n = 9 ( T9x9 ):
1 1.1954519748687744 0.39188057184219360 0.12402343750000000 0.39188057184219360 0.24804687500000000 0.10616630315780640 0.12402343750000000 0.10616630315780640 5.2594900131225586E-002 2 0.39188057184219360 1.3194754123687744 0.39188057184219360 0.24804687500000000 0.49804687500000000 0.24804687500000000 0.10616630315780640 0.17661833763122559 0.10616630315780640 3 0.12402343750000000 0.39188057184219360 1.1954519748687744 0.10616630315780640 0.24804687500000000 0.39188057184219360 5.2594900131225586E-002 0.10616630315780640 0.12402343750000000 4 0.39188057184219360 0.24804687500000000 0.10616630315780640 1.3194754123687744 0.49804687500000000 0.17661833763122559 0.39188057184219360 0.24804687500000000 0.10616630315780640 5 0.24804687500000000 0.49804687500000000 0.24804687500000000 0.49804687500000000 1.4960937500000000 0.49804687500000000 0.24804687500000000 0.49804687500000000 0.24804687500000000 6 0.10616630315780640 0.24804687500000000 0.39188057184219360 0.17661833763122559 0.49804687500000000 1.3194754123687744 0.10616630315780640 0.24804687500000000 0.39188057184219360 7 0.12402343750000000 0.10616630315780640 5.2594900131225586E-002 0.39188057184219360 0.24804687500000000 0.10616630315780640 1.1954519748687744 0.39188057184219360 0.12402343750000000 8 0.10616630315780640 0.17661833763122559 0.10616630315780640 0.24804687500000000 0.49804687500000000 0.24804687500000000 0.39188057184219360 1.3194754123687744 0.39188057184219360 9 5.2594900131225586E-002 0.10616630315780640 0.12402343750000000 0.10616630315780640 0.24804687500000000 0.39188057184219360 0.12402343750000000 0.39188057184219360 1.1954519748687744
It is straightforward to show that the solution vector x is given by:
x = T . b . . . (2)
T is the Cairo tensor.
Equation 2 holds for all values of a and b.
Consider the example proposed by author and mathematics professor John Mathews [1]:
b = [100, 20, 20, 80, 0, 0, 260, 180, 180] T . . . (3)
Mathews classically solved the resulting system of nine linear algebraic equations by the Gaussian elimination method, using a more efficient scheme. He extended the tridiagonal algorithm to the more sophisticated pentadiagonal algorithm for his arbitrarily chosen vector b (boundary conditions). This gives the solution vector:
U=[55,7143,43,2143,27,1429,79,6429,70,0000,45,3571,112,357,111,786,84,2857]T.. . . (4)
The BC vector of the proposed statistical solution corresponding to equation 5 is now simply rewritten: [100/4, 20/4, 20/4, 20/4, 80/4, 0, 0,260/4, 180/4, 180/4] T.. . . . . (5) The calculated transfer matrix D = E-I, represented by T1, can be multiplied by the BC vector (b). We thus obtain: U = [55.7132187 43.2126846 27.1417885 79.6412506 69.997863845.3555412112.856079 111.784111 84.2846451] T. . . . . (6).
Comparing the numerical results of the proposed physical statistical solution (Eq. 6) with the mathematical results of Mathews (Eq. 4), we observe a striking accuracy.
Third example
=========
The question arises: Is this procedure also valid for quantum mechanical systems?
In short, the answer is yes.
For the same structure as in Example 2, with a source or boundary condition at the center of mass CM (node 5), and using the Cairo tensor from Equation 2, the 2D solution is precisely triangular, as follows:
[121 232 121 ]
What is the solution for the square of the wave function ψ^2= ψ . ψ* for a quantum particle in a closed box?
The final step is to calculate the square root of the above solution, which should describe ψ itself.
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1- I. Abbas, A Fundamental Correction for Modeling Nature via Numerical Statistical Theory: The Image Becomes straight.
The short answer to your question is no - mathematics is not a branch of physics. Mathematics is a foundation of physics.
Your example of the 4-dimensional hypercube is incorrect. I've never seen 8a^4 as the volume of a 4-dimensional hypercube. Mathematics certainly doesn't suggest this. Mathematics has the volume of a 4-D hypercube with side length a as a^4. Can you give a citation that shows the volume is 8a^4?
With your second example, where is a in equation 2?
No, Mathematics is the mother of all branches of Science also Engineering.
Please take a look at Google search:
[ Yes, physics can generate new mathematics; the history of science shows that physical problems, such as understanding motion or heat flow, have directly inspired and led to the development of entirely new mathematical concepts like calculus and Fourier analysis. While math is a tool used by physics, physical insights can also prompt mathematicians to create the abstract tools and systems needed to describe and analyze natural phenomena, demonstrating a symbiotic and ongoing relationship between the two fields.
How Physics Generates Math
- Solving Physical Problems:When physicists encounter problems, like calculating the motion of objects or understanding heat transfer, they may find that existing mathematical tools are insufficient.
- Developing New Concepts:This inadequacy motivates the creation of new mathematical frameworks and techniques. For example, Isaac Newton and Gottfried Wilhelm Leibniz developed calculus to understand motion, and Joseph Fourier developed trigonometric series to study heat flow.
- Unlocking New Insights:Physics can provide the "why" behind abstract mathematical concepts, inspiring mathematicians and helping them to see the relevance of their work in the real world.
The Relationship Between Physics and Math
- Math as a Tool for Physics:Physics relies heavily on mathematics to describe and quantify natural phenomena, model relationships, and make predictions.
- Physics as a Source of Mathematical Inspiration:The interaction is not one-sided. Insights from physics can inspire mathematical breakthroughs, showing an "unreasonable effectiveness" of math in describing the physical world and, conversely, of the physical world in generating new mathematics.]
Dear, the entire Universe or COSMOS is run based on Physical Principles and NOT on Mathematical rules. It implies that the subject of Mathematics is part and parcel of Physics.
I think Physics is a branch of Mathematics, in the physical world. And also Mathematics is a branch of PHYLOSOPHY.
We offers an alternative answer below to illustrate that mathematics is, in a sense, a branch of physics.
The revolutionary physical theory of the Cairo techniques generates new mathematics, such that Schrödinger's partial differential equation (PDE) is better described by its complex transition matrix/tensor Q [from physics], as follows:
We assume that neither E. Schrödinger nor N. Bohr understood physics and that the classical Schrödinger equation, interpreted according to Bohr/Copenhagen, is a disaster.
A mathematician who does not master the universal laws of physics is not qualified to conduct mathematical research.
The author proposes another new mathematics such that the Schrödinger partial differential equation (PDE) is best described by its complex transition matrix/tensor Q, as follows:
Q=√B
where B is the transition matrix well known in the theory of Cairo techniques.
For example, the transfer matrix D(N)=Q+Q +Q ..etc for 8 equidistant nodes in a 3D cube is expressed as follows:
D(N)=
((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840
((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840
((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840
((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840
((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840
((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840
((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840
((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5-(63-45*i)*6^0.5+32)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5-(21+15*i)*6^0.5+64)/840 ((105-35*i)*2^0.5+(21-15*i)*6^0.5+176)/840 ((105+35*i)*2^0.5+(63+45*i)*6^0.5+928)/840
By multiplying D(N) by the vector [11111111], the Rayleigh constant can easily be calculated as 3.414214, and then π = 3.14159.
===
Note: If you are unfamiliar with the universal laws of physics, please stop reading.
This answer is not for you.
Historically: Some physicists (like Galileo, Dirac, Feynman, Tegmark) have argued that physics is the "realization" of mathematics in nature. From that view, mathematics could be seen as a tool emerging from describing physics.
Philosophically: Most mathematicians and philosophers of science argue the opposite — mathematics is more general and abstract, while physics is an application of mathematics to the natural world.
Modern consensus: Mathematics is not a branch of physics. Rather, it is an independent discipline that provides the formal language and structures that physics uses.
However, in frontier research (like string theory, quantum gravity, or topology in condensed matter), the boundary between math and physics becomes very blurred.
Mathematics seeks universality: truths that hold regardless of the physical world.
Physics is empirical: it only keeps the mathematics that “fits” with observed reality.
So from the mathematicians’ side, it’s natural to reject the claim that math is “a product of physics,” because mathematics clearly extends far beyond what physics has yet explored.
I use mathematics to develop statistical methods for econometrics: there is no relation to physics.
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