Nature was written in the language of mathematics. This statement from Galileo illustrates the importance of mathematics in understanding...
Nature was written in the language of mathematics. This statement from Galileo illustrates the importance of mathematics in understanding nature. Real big discoveries almost always include math. New knowledge of nature is really a discovery if we can describe the phenomenon with mathematical formulas. And when it comes to putting nature at our service, we almost always use math to design. Mathematics plays a central role in nature. Nature was really written in the language of mathematics. But why is this so? Why can mathematics play such a central role? Why does mathematics work in our world at all?
In the process of learning about nature, scientists have wondered countless times about the role of mathematics. Eugene Wigner's article on this topic, quoted in many places, entitled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", circles the topic without a clear conclusion.
There are also ideas that the universe is actually a mathematical structure. Max Tegmark deals with this idea in his book: The Mathematical Universe.
In fact, mathematics is, for some reason, somehow the essence of nature. Why this might be the case, to arrive at an implicit or real conclusion, is to follow Eugene Wigner's line of thought.
What is Mathematics?
Mathematics is the science of quantities and operations with quantities. Mathematical operations form one or more quantities into another quantity. Operations can also assign properties, qualities interpreted in mathematics to quantities. Mathematics looks for relationships between these qualities, identifies similarities, shared common features, and groups them based on these features. Mathematics creates structures from quantities, groups created by operations, and qualities, and examines the properties of those structures.
This definition also applies to geometry, as geometry can be handled through quantities and operations.
Why does math have any structure at all? Why do structures exist in mathematics? Why mathematics can create structures?
The basis of mathematics consists of discrete, separated elements of the same property, consists of numbers. No matter how close two non-identical numbers are to each other, they are different, and if two numbers are the same, they are the same in all their properties. Based on this basis, operations combine quantities, create new properties and groups. The operations apply equally to the underlying quantities. Identical operations mean the same procedures for all the quantities constituting the base. Mathematical structures are created by mathematical operations. Structures are made up of groups of quantities associated with operations having the same properties, properties of the resulting structures, and their interrelated relationships.
An important feature of mathematics is that the resulting mathematical structures may have new, emergent properties that are not carried by the elements of the structure.
As a special case, some operations may produce the same quantity on different quantities of the base. Such operations create so-called symmetry in groups of given quantities. Symmetry operations are special because they assign different quantities to the same quantity, meaning that the quantity remains the same during the operation.
Mathematics is an abstract science, an idealized process. Mathematics does not deal with the quality of quantities. In mathematics, quality, physical reality, plays no role. One, as a quantity, can refer to anything, and one and one can certainly be two, whether or not the two things can be added together, whether there is a sense in adding the two in reality or not.
Mathematics is a system of rules applied to quantities, irrespective of what the quantity relates to. Therefore, mathematics is a universal science. The content, meaning, and rules of mathematics are the same everywhere in our universe, even if our universe is a multiverse.
What is physics (and practically all of natural science)?
Physics is a description of reality, of the existing world, of nature. Through physics, we get to know the elements, properties, relationships, and interactions of reality. Physics describes the real world, establishes facts about existing things, observes its properties. It seeks out, defines, and describes the building blocks of the world, how they interact, how they interact with each other, how structures are built by interaction.
Physics examines how increasingly complex structures are built up and determines the properties, interactions, rules, and regularities of these structures.
Physics basically deals with reality, but we can extend physics to unexperienced reality. For example, physics can deal with the multiverse, but in this case, also the goal is to get to know the reality. We invent the multiverse because we think it is part of reality.
Physics can deal with alternative reality, but in this case, too, physics looks for a realistic reality, physics assumes that these alternative realities can exist.
An important feature of the reality is that the structures created by the interactions can have new, emergent properties that are not carried by the building blocks.
Comparison of mathematics and physics - similarities and differences
Mathematics is abstract, physics is a concrete science
Mathematics deals with quantities, numbers. Everything has a quantity, but quantity is not an intrinsic property of things, but an extrinsic property, so it can be detached, abstracted from a given thing, so mathematics can be handled abstractly from material quality.
Mathematics expands quantity, using zero, negative, and even more abstract quantities, such as irrational quantities.
Physics is a concrete science, always referring to existing quality, thing. Physics cannot be abstracted from the reality it discusses.
Mathematics is universal, physics is a specific science
Because math is not related to the quality of things, it doesn't depend on it. Mathematics can be applied to any material quality. Of course, not all mathematical operations can be interpreted, and therefore not all mathematical operations can be used for different material qualities, but this is not a limitation of mathematics, but a feature of our material world.
The statements of physics are always related to the given material quality and cannot be separated from the material quality. Physics is specific. Of course, certain physical laws may apply to multiple material systems, but this does not mean the universality of physics, but it is a feature of our material world.
Mathematics is a purely theoretical science, physics is based on observation.
Since mathematics is not concerned with material quality, it is theoretically manageable, formatted, and developed without taking into account our material world. The rules and limitations of its operation are basic logic, obvious truths. Since mathematics is not constrained by reality, it is not an uncontradictory science (Gödel's recognition).
The basis and limitation of physics is observation, the existing world. Physics can be improved theoretically, even with the tools of mathematics - in which case we are amazed at the role of mathematics in nature - yet the finest theory, the most logical conclusion, can prove to be wrong in physics - as it often happens - if it does not correspond to reality.
Not all interpretable operations of mathematics are interpretable operations of reality. This property of reality also ensures that reality creates a system free of contradictions. If a theoretical inference leads to contradiction in physics, then that theoretical path certainly cannot be valid in reality.
The recognition of mathematical knowledge is a theoretical conclusion, but the possibility of contradiction cannot be ruled out.
Physics performs essentially the same task as mathematics. Both deal with systems of elements. Mathematics performs mathematical operations on elements, physics examines existing interactions. Both deal with the properties of new structures created by operations-interactions.
Physics is more than mathematics because it deals with the quality too, and this is why it less also. Physics is more limited than mathematics in that reality is a subset of all possible possibilities.
An interesting and important feature of both sciences is that both are open. It is common in both sciences that operations and interactions create new systems with new properties. The theoretical limit of physics is determined by the finiteness of the materialized reality, whereas mathematics has no such limit.
What is the relationship between mathematics and physics?
Mathematics is the tool of physical knowledge. Since mathematics deals with quantities, a characteristic of reality that is independent of the quality of reality, mathematical operations are valid in physics.
Conversely, the statement is not valid. Physics is not a tool for understanding mathematics. The real-world limits the validity of mathematics. For a real material system, not all mathematical operations are meaningful or make sense. Physics, reality, limits the mathematical operations that apply to a given material system.
An essential feature of physics is that the mathematical operations that apply to a given physical system are universally valid. They operate regardless of location, time, and other circumstances. These are symmetries of our physical reality. These symmetries exist because there is valid symmetry in the mathematical description of the given physical interactions.
The relationship between physics and mathematics is a one-way direction. Physics always follows the (actually valid) rules of mathematics, but physics, reality, is not the definitive constraint on mathematics. The limitations of mathematics are pure logic (if the steps of a proof are true then its consequence and the result of the proof are true), its starting rules are obvious or abstract truths (for example, if two quantities are different then they are not the same or two lines which do not touch have no common point).
The path discovering the reality, the validity of the steps of a conclusion is based (besides, instead of pure logic) on the inaccurate method of observation, and abstract truths are often unintelligible in reality (two apples can be different or can be the same, and in reality, there is no infinitely long, infinitely thin structure).
The real world, physics is based on interactions, interactions are operations of the real world. Interactions govern how the real-world works. Physical interactions have specific rules. Since interactions also mean quantities, physical interactions can be described using mathematical tools, but because interactions are specific, they have specific characteristics, so only certain mathematical tools can be used, only certain mathematical tools are valid for the given physical interaction. Discovering these specialties is the purpose of physics as a science, and mathematics provides the tools for this.
Why does mathematics work in discovering reality?
Because the reality is based on quantities, mathematics is naturally applicable discovering the knowledge of the existing. The specificity of existing interactions that actually valid in reality related to a physical quantity governs and determines which mathematical operations can be applied to a given material system. For example, in quantum mechanics, it is necessary to use complex numbers. Why? Because the system, structure, symmetry of the interactions of quantum mechanics has comparable physical symmetry as the operations of complex numbers, the structure of that mathematical tool. In other areas of physics, similar correspondences apply to other mathematical tools.
The progression of physics is based on observation. However, once we are able to get an approximate view of a given physical system and its interactions, physics can evolve theoretically, using the tools of logic, but any new deduction can only be valid if it is supported by observation, so when the reality is consistent with the deduction.
What makes mathematics the language of physics?
If mathematics is so idealized and so independent of reality, how can it still be so fundamental, valid, and decisive in physics?
Everything that has been stated so far in mathematics and physics demonstrates the close relationship between mathematics and physics. However, all the statements so far, while essential, do not point to the fundamental relationship. The basic question remains. Why is it one and one equals two in physics also? Why is that one and one always two everywhere in the universe? What makes mathematics the language of physics? Why does reality follow the rules of mathematics? Why we can use mathematical operations and rules for physical quantities?
There seems to be an exception to this rule. There seems to be an exception in natural laws where mathematics based on logic does not apply. For example, adding speeds. We know that, the special theory of relativity has shown, and reality has proved, that what should be valid under the rules of basic logic, a simple summary of speeds is not a valid operation of reality. It seems like reality did not obey mathematics. As if it were an exception to the universality of mathematics. In fact, in this case, too, the validity of mathematics is not compromised. The rules and operations of mathematics also apply to the addition of speeds, but the reality, our reality, is that velocities cannot be summed up by the mathematical operation of addition. Adding together speeds is subject to a more complex mathematical operation which, to be specific to our reality, also contains a natural constant. The mathematical operations for the addition of velocities, the simple operation of summary of mathematics is just an approximation of the reality, we can describe an even more precise relation. When adding velocities, mathematics was not inaccurate, but reality does not use mathematics as a simple addition but rather a more complex mathematical formula. The mathematics, the mathematical formula and the general mathematical operations used in it are valid.
The relationship between physics and mathematics must be very close, since mathematics, its valid form has unlimited validity in physics. We may have to rewrite the mathematical formulas of physics from time to time, but never because the principles of mathematics prove to be flawed, but because it turns out that our idea of reality, based on experience and observation, was inaccurate. Mathematics is flawless, our concept of reality, to which we applied mathematics to describe its rules, is inaccurate.
However, the basic question still remains. Why do the operations of mathematics apply unlimitedly, anytime, anywhere in physical reality?
The question seems unanswered, but perhaps there is an approach that can bring us closer to the solution. Turn the question over. Don't look for why math works in nature. Let's try to answer how nature should be built where mathematics has the role that we can experience.
What should be the reality where mathematics is the basic form of the description, where the basic rules are the rules of mathematics?
As stated above, mathematics exists, has a structure, operations are universally valid in mathematics because - and now a quote follows:
"The basis of mathematics consists of discrete, separated elements of the same property, consists of numbers. No matter how close two non-identical numbers are to each other, they are different, and if two numbers are the same, they are the same in all their properties. Based on this basis, operations combine quantities, create new properties and groups. The operations apply equally to the underlying quantities. Identical operations mean the same procedures for all the quantities constituting the base. Mathematical structures are created by mathematical operations. Structures are made up of groups of quantities associated with operations having the same properties, properties of the resulting structures, and their interrelated relationships."
Mathematics is a science, it can work, and mathematical operations are universally valid within mathematics because mathematics is based on discrete, clearly separated elements of the same property. Physical interactions can be compared to mathematical operations. So, if the reality is constituted by discrete, clearly separated elements of the same property, the reality that is built on this basis may be the subject of mathematical operations. That must be our reality.
The Consequence of the Validity of Mathematics - Conclusion
Since the operations of mathematics are valid everywhere in physical reality, it must be true for every part of our physical reality that the physical reality is made up of clearly separated elements of the same property. It must be true for the empty space too. For mathematics to be generally applicable to physical reality, the basis of empty space must also be built on, must consist of discrete, clearly separated elements of the same property.
Mathematics can work with quantities infinitely close to zero and infinitely close to each other. That's why math is universal. Nature is probably not like that. Reality does not like the infinite. Our reality probably has the smallest unit and probably also has the smallest distance between the units. Above these limits, mathematical operations prevail in the physical reality constructed in the described way.
Following this logic, our world, where mathematics is valid, must be like that. But is this our world? Does experience, observation support this logical conclusion? The science of the smallest, quantum mechanics, does not contradict this. Quantum mechanics has already recognized the smallest amount, it can be attributed to the Planck constant, and in quantum mechanics, the void is not empty but behaves like ubiquitous quantum foam.
Our concept of the smallest reality does not contradict the theoretical conclusion of the relationship between mathematics and physics. However, the theory of relativity, the science of large quantities, excluded the presence of an everywhere existing substance, the existence of the aether. Rather, the truth is that relativity does not exclude the existence of the aether, but does not see the presence of the aether and does not need the existence of the aether.
However, the general validity of mathematics suggests that there is a basic structure nature built on. Aether, as a basic structure, can exist in physical reality, even if not in the way that relativity initially assumes.
The grid model sketched in earlier thoughts, a model describing physical reality as a grid of identical particles, can also incorporate relativity theory, the theory that is supported by observations. Can the grid model be a model of reality? Maybe not. However, the notion of the relationship between mathematics and physics supports the validity of the grid model.
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