It is true that mathematics is a product of the universal laws of physics.
A mathematician who does not master the universal laws of physics is not qualified to master the laws of mathematics.
A mathematical law that contradicts a single law of the universal laws of physics would be false.
Mathematics is not a product of physics. It is an independent logical framework, though it serves as the language of physics. While physics inspires many mathematical ideas, mathematics also develops concepts with no direct physical application. Their relationship is one of mutual enrichment, not dependence.
A Refutation of the Claim that Mathematics is a Product of Physics
The statement "mathematics is a product of the universal laws of physics" is a strong claim that can be challenged by considering the historical development and abstract nature of mathematics. While there is a deep and undeniable synergy between mathematics and physics, to claim that one is a product of the other is to misrepresent the foundational and creative aspects of both disciplines.
1. The Existence of Abstract Concepts and Idealizations
Mathematics frequently operates with concepts that have no direct, observable physical equivalent, yet remain entirely consistent within their own logical framework. The example of the limit of 1/x as x→0 approaching infinity is a prime case. In the physical universe, we can discuss immense numbers of particles or vast distances, but the concept of a true, mathematical infinity is an abstraction. It is an ideal tool that allows physicists to make useful approximations in fields like thermodynamics or cosmology, but the tool itself is not a physical law. Mathematical infinity is a conceptual construct, not a physical object, which demonstrates that mathematics can create its own valid realities independent of the physical world.
2. The Development of Fields with No Prior Physical Basis
A strong counter-example is the development of non-Euclidean geometries in the 19th century by mathematicians such as Lobachevsky, Bolyai, and Riemann. These geometries were created as purely logical exercises, exploring what would happen if the parallel postulate of Euclidean geometry were abandoned. At the time, they were considered intellectual curiosities with no apparent connection to the physical world, which was thought to be perfectly described by Euclidean geometry. Over half a century later, Albert Einstein utilized these very mathematical structures to formulate his theory of general relativity, showing that the geometry of spacetime is, in fact, non-Euclidean. This historical sequence clearly demonstrates that mathematical concepts can be developed for their own internal consistency, predating any known physical application.
3. The Utility of Non-Physical Numbers
Consider the set of imaginary numbers, most famously represented by the constant i=−1. While there is no real-world, measurable quantity that corresponds to i, it is a cornerstone of advanced mathematics and physics. Complex numbers are absolutely essential for describing phenomena in quantum mechanics, electrical engineering, and signal processing. The very idea that a number could be the square root of a negative value is a purely mathematical invention, yet it provides an indispensable language for describing physical reality. If mathematics were simply a product of physics, a concept like i would be an impossible contradiction. Instead, it proves that the rules of mathematics are broader than the laws of our universe.
In conclusion, while physics has often provided inspiration and problems for mathematicians to solve, mathematics itself is a self-contained, creative, and abstract discipline. It provides a universal language for describing reality, but it is not constrained by it. Mathematical laws do not need to be validated by the physical world to be true within their own system, and this is why they are so effective at pushing the boundaries of scientific inquiry.
How can some mathematics be possible in physics, but not in mathematics?
It's simple.
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.
No, it’s the other way round—physics is a branch of mathematics.
Physics is ultimately about understanding the natural world—matter, energy, motion, forces. To do that, it needs a precise language, and that language is mathematics. Physics doesn’t invent mathematics; rather, it borrows existing mathematical structures (like calculus, linear algebra, differential equations, group theory) to describe natural laws in a consistent way.
Mathematics, on the other hand, exists independently of physics. A mathematician can study abstract concepts like prime numbers, topology, or category theory without any concern for physical reality. Many such ideas later turn out to be useful in physics, but they weren’t created for physics in the first place.
So, while mathematics can stand entirely on its own, physics cannot be formulated rigorously without mathematics. In that sense, physics can be viewed as a branch or application of mathematics, rather than the other way around
About the tesseract volume, the formula is the same in physics and mathematics. The "8" comes from the *surface* formula, *not* volume.
Answer II
We demonstrate once again how many mathematical solutions are possible in physics, but not in mathematics.
This time, we analyze a problem more related to physics than to 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)
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.
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1- I. Abbas, A fundamental correction for modeling nature via numerical statistical theory: the image becomes upright.
No, Mathematics is the mother of all branches of Science also Engineering. Mathematics is the base of all study.
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.]
To quote J. J. Sylvester, “Just like music, math is built on a very rigidly defined set of rules and our mathematical constructions exhibit an infinite amount of creativity.” Or as mathematicians will often say, mathematics is the music of reason.
That mathematics can be used to describe physical phenomena is irrelevant. It can also be used to describe biological phenomena, economic phenomena, etc.
Some will tell you mathematics is about numbers, but it is not. It is about the idea of numbers as there is no real preferred number system. The integers, reals, complex numbers, algebraic number fields. Then modular numbers. Abstracted further to algebraic groups such as elliptic curves where points on a on an elliptic curve can be combined to another point on the curve that behave as addition.
Physics, like all other sciences, is an experimental science where conjectures to describe observed phenomena are developed and then validated or falsified through experiments. To quote Richard Feynman, "It does't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it is wrong.
An interesting read.
https://magazine.krieger.jhu.edu/fall-2018-v16n1/the-art-of-mathematics/
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