# How is it that mathematics describe reality so well?

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raphalbatros
Humans created that tool, that language, that consists of axioms and their implications. Mathematics do a good job of communicating the behavior of physical phenomena. In some instances, pure mathematical areas have been found to describe some aspect of reality, only years after having been studied by mathematicians.
Mathematics, as a concept, is a closed system where deduction is the key method of reasonning.

My question:
How is it that mathematics and and the physical reality agree with each other ?

A proper definition of those two concepts might have to be established if we are to proceed to answer my question.

Mentor

Swapnil Das, MexChemE and ProfuselyQuarky
Hey raphalbatros.

Basically because it turns out as mathematics add abstraction and make it consistent - they find a system that corresponds to what is observed/experienced in the physical realm.

The push for abstraction is often a long one and it is often met with resistance - but when breakthroughs of insight (and with effort) are made then this happens.

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2021 Award
The Unreasonable Effectiveness of Mathematics in the Natural Sciences is a nice article about it (it even got its own wikipedia page).

Basically because it turns out as mathematics add abstraction and make it consistent - they find a system that corresponds to what is observed/experienced in the physical realm.

I first wanted to say something like: "IMO there are two universal languages, math and music. One describes what's going on inside of us, and the other one what's going on outside." Then I tripped over the adjective.

Imagine a category defined by all languages which describe a certain physical phenomena as objects and the translation between them as morphisms. Does mathematics have a universal property in this category? My hypothesis is "Yes, it has." because there are no constraints on mathematical concepts. If they don't exist already, the may be added. And not even an overall consistency is required, only local.

protonsarecool
ogg
Math is NOT "a closed system" !

Demystifier
ogg
Also, fwiw, I found the Nova program so assumption and error laden that I stopped watching it at ~16 minutes. The alarming "information" up to that point were
1. The false implication that there are no 4 petal flowers. I'm looking at one right now. (Interesting fact: no 4 petal flowered plants are native to North America)
2. The silly claim that circles don't 'exist' when a pin is dropped onto a ruled piece of paper. Uh, the pin's location can be characterized by its xy position and Θ of (circular) rotation.
3. The even sillier claim that some character symbols (on a wall) ARE math. They represent math if and only if those reading them have an enormous amount of background knowledge, specific knowledge about those symbols' relationships to an huge body of other knowledge.
4. The show drones on and on about the Western System of Music notes, which was partially created to enshrine math relationships, AND which is only one of many systems, where in the others notes don't occur as simple integer fractions of one-another. It's like putting only even numbers into a hat, and marveling when you randomly remove them and find that none are odd.
5. There is no math which EXACTLY fits reality. Our best theories (the Standard Model of Particle Physics) has not even been proven (despite great effort) to be consistent (it's a Millennium Prize goal). To be specific, ask a mathematician, engineer or physicist to tell you exactly where a ball you roll down a hill will come to rest. How far off do you think they will be? nanometers? millimeters? inches? meters? Almost certainly meters. Or consider the probabilistic nature of the sub-microscopic world, and then define what a "random" number is. ...
6. Your claim is like someone claiming that epicycles are amazingly and puzzling accurate depiction of planetary orbits. Well, only if you ignore their inaccuracies.

raphalbatros
This NOVA documentary doesn't answers my question at all. It seems to have been made to capture the attention of some people who are not familiar with what is are mathematics. ogg makes some good points about the video in question.

I have already looked at that article in the past (The Unreasonable Effectiveness of Mathematics in the Natural Sciences), but I stopped at mid-lecture, I don't think it is answering my question.

ogg
Before arguing over the fact that math is a closed system, or not, I would like to elaborate on what I mean by "closed system". I might not be using this expression the right way, but I mean a system which has a finite number of axioms, from which you can deduce things. I don't mean a system that can't be changed, or a system that has a finite number of implications.
In your points 5 and 6, you claim that math doesn't exactly fits our reality, but you give an explanation for why current theories in physic doesn't exactly fits our reality. For what I know, math has done a perfect job, so well, to communicate the behavior of physical phenomena. What do you think about that ?

write4u
Also, fwiw, I found the Nova program so assumption and error laden that I stopped watching it at ~16 minutes. The alarming "information" up to that point were
1. The false implication that there are no 4 petal flowers. I'm looking at one right now. (Interesting fact: no 4 petal flowered plants are native to North America)
In the presentation no one suggests that there are no 4 petal flowers. The suggestion is that in some species of flowers and other natural constructs such as spiral galaxies, the Fibonacci sequence seems an efficient natural (mathematical) method of making petals and regular spirals. The same goes for *fractal* iterations seen in many plants, indeed throughout the universe.
2. The silly claim that circles don't 'exist' when a pin is dropped onto a ruled piece of paper. Uh, the pin's location can be characterized by its xy position and Θ of (circular) rotation.
That may be true, but I believe the point was that one can drop the pin randomly and calculate Pi, without invoking relationship between the diameter and circumference of a circle. I saw no circle, I saw no diameter, only a random action with straight objects.
3. The even sillier claim that some character symbols (on a wall) ARE math. They represent math if and only if those reading them have an enormous amount of background knowledge, specific knowledge about those symbols' relationships to an huge body of other knowledge.
Is that not why we have the scientific disciplines of mathematics and physics?
4. The show drones on and on about the Western System of Music notes, which was partially created to enshrine math relationships, AND which is only one of many systems, where in the others notes don't occur as simple integer fractions of one-another. It's like putting only even numbers into a hat, and marveling when you randomly remove them and find that none are odd.
Obviously you are not a musician. Ever heard the term *discord* or "noise*? Of course the clip also showed that Pi occurs in the *wave function* and in *meanderings* of rivers. Seems to me that a persuasive case was made that Pi is much more than the relationship between a circle and its diameter.
5. There is no math which EXACTLY fits reality. Our best theories (the Standard Model of Particle Physics) has not even been proven (despite great effort) to be consistent (it's a Millennium Prize goal). To be specific, ask a mathematician, engineer or physicist to tell you exactly where a ball you roll down a hill will come to rest. How far off do you think they will be? nanometers? millimeters? inches? meters? Almost certainly meters. Or consider the probabilistic nature of the sub-microscopic world, and then define what a "random" number is.
As I understand it, the law of falling bodies is a constant and works for any length (at least in a vacuum). The fact that we may not be able to measure correctly has nothing to do with the underlying exactness of the Law of falling bodies.
6. Your claim is like someone claiming that epicycles are amazingly and puzzling accurate depiction of planetary orbits. Well, only if you ignore their inaccuracies.
Are those inaccuracies random and spontaneous or are they caused by external or internal forces which can be mathematically calculated ?

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Humans created that tool, that language, that consists of axioms and their implications. Mathematics do a good job of communicating the behavior of physical phenomena. In some instances, pure mathematical areas have been found to describe some aspect of reality, only years after having been studied by mathematicians.
Mathematics, as a concept, is a closed system where deduction is the key method of reasonning.

My question:
How is it that mathematics and and the physical reality agree with each other ?

A proper definition of those two concepts might have to be established if we are to proceed to answer my question.
Simply put: Someone chose the part of mathematics that could be used to describe a certain phenomenon. Lots of mathematics exist that have no connection to the physical world.

Geofleur
write4u
Simply put: Someone chose the part of mathematics that could be used to describe a certain phenomenon. Lots of mathematics exist that have no connection to the physical world.
To me that is a very interesting statement. Do you mean theoretical mathematics or certain mathematical hierarchical constants which we have not discovered yet, or are still dormant until explicated in the physical world ? Potentials?

raphalbatros
write4u have a point. the arguments 1 and 2 made by ogg are invalid I think. They claim the documentary discarded some things (like 4 petals flowers and the presence of circles in an experiment), but in fact, it just did not mention these.
On the other hand, the fact you don't see a circle or a diameter does not mean there is not one. I would even argue that when you find pi somewhere, there is always the presence of the ratio between a circle's diameter and its conference hidden somewhere.

Svein, I understand what you are saying, but it does not adress my question. Let me rephrase it.
How is it that some parts of mathematics and the physical reality agree with each other ?
Basically, I want to dive a bit further into the reasonning.

To me that is a very interesting statement. Do you mean theoretical mathematics or certain mathematical hierarchical constants which we have not discovered yet, or are still dormant until explicated in the physical world ? Potentials?
Mathematical fields looking for an application (off the top of my head, it may not be all correct):
• Parts of number theory - very theoretical
• Advanced mathematical logic (think Gödel's theorem)
• Function algebras
• Advanced topology (we know the mathematical definition of a Klein bottle, but we cannot create one)
• ...
Mathematical constants we do not know enough about:

(see https://en.wikipedia.org/wiki/Euler–Mascheroni_constant )

Gold Member
To me that is a very interesting statement. Do you mean theoretical mathematics or certain mathematical hierarchical constants which we have not discovered yet, or are still dormant until explicated in the physical world ? Potentials?
There are entire fields of mathematics that did not arise as language devised to describe empirical observation.

Gold Member
IMO one can never know why mathematics can be used to explain physical phenomena. This is the great mystery.

I think your question is essentially metaphysical not scientific. Science by itself can never answer why.

I would point out though that the view that mathematics is merely language used to describe empirical phenomena - a point of view that seems almost universal on the Physics Forums - is itself metaphysical and is not the only point of view held historically by scientists, theologians, mathematicians, and philosophers.

Here is a quote from the mathematician David Hilbert that I found as footnote #18 on the Wikipedia page that reviews Hilbert's life and career.

"Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs." David Hilbert, Die Grundlagen der Mathematik, Hilbert's program, 22C:096, University of Iowa.

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Mentor
2021 Award
There are entire fields of mathematics that did not arise as language devised to describe empirical observation.

I think your question is essentially metaphysical not scientific. Science by itself can never answer why.

The idea, comparison resp. of a language might be stressed a little further.
English is a language in which one can describe all empirical observations. Nevertheless it is not established to do in the first place. There is a huge variety of English vocabulary that isn't meant to describe nature. And if needed, new words will be introduced to complete its ability to do so. To the extend discussed here, mathematics serves a similar purpose. In this regard the only difference between English and mathematics is that nobody questions English.

I know well that this view is a reduction, but IMO it answers the question in the thread title: Mathematics can be seen as a developing language which happens to be useful to also describe empirical phenomena.

lavinia
Gold Member
I know well that this view is a reduction, but IMO it answers the question in the thread title: Mathematics can be seen as a developing language which happens to be useful to also describe empirical phenomena.

A book I am now reading called "The Theological Origins of Modernity" by Michael Gillespie talks about the Problem of Universals which is also summarized in this Wikipedia article.

https://en.wikipedia.org/wiki/Problem_of_universals#Realism

Realism seems to be the view that universals such as mathematical concepts - but also many other things - are real and that empirical objects are particular instances that exemplify them. The form and laws of Nature follow from universals more or less by deduction. This view is attributed to both Platonism and Aristotelianism. A third view called Nominalism denies the reality of universals and rather says that they are "signs" or methods of description. In this view mathematics and more general categories are just language used to explain Nature. A fourth view is that universals are intrinsic to thought itself. This is a more modern point of view which I think comes from Kant.

Interestingly, Leonard Susskind's Youtube Lectures on the Special Theory of Relativity start with the universal that the speed of light is the same in every inertial reference frame and from it, deduces the empirical consequences. This seems to vindicate the Realist point of view.

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fresh_42
"Mathematics can never prove anything. No mathematics has any content. All any mathematics can do is -- sometimes -- turn out to be useful in describing some aspects of our so-called 'physical universe'. That is a bonus; most forms of mathematics are as meaning-free as chess."

Robert A. Heinlein

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2021 Award
A third view is called Nominalism and this denies the reality of universals and rather says that they are "signs" or methods of description that humans use to describe things. This is the view that mathematics and more general categories are just a language used by humans to explain Nature.

"In metaphysics, the problem of universals refers to the question of whether properties exist, and if so, what they are. Properties are qualities or relations that two or more entities have in common. The various kinds of properties, such as qualities and relations are referred to as universals."
(Wikipedia, s.a.)

The metaphysical and categorial definition of universals are probably not by chance very similar. Following Nominalism this implies that categorial objects (knowingly) lack realism? This is a statement I understand although I'm not sharing it. I would associate Wittgenstein here.
The question that comes up to me is how to categorize the mathematical or informatic theory of formal languages, which indeed has real applications nowadays. Isn't it an example how even metaphysics fit in areas of mathematical research? Not to speak about Gödel's results.
Most of it happened after Wittgenstein's lifetime. Nevertheless, to me these are valid arguments that the distinction between language and reality is artificial. At least it means one has to be extraordinary precise with the definition of meta-levels. (As to my knowledge philosophers still haven't come up with a satisfactory description of what is meant by reality. A term which probably each individual has to find an answer to by itself.)

Mentor
2021 Award
"Mathematics can never prove anything. No mathematics has any content. All any mathematics can do is -- sometimes -- turn out to be useful in describing some aspects of our so-called 'physical universe'. That is a bonus; most forms of mathematics are as meaning-free as chess."

Robert A. Heinlein
Strange. That is mainly what I think about physics.

anorlunda and lavinia
Gold Member
My question:
How is it that mathematics and and the physical reality agree with each other ?
Mathematics only agrees with the physical reality that we can observe. As soon as someone observes something else, we often need to change - or at least adjust - the math.

Mathematics is basically a game where we create a problem that needs to be solved by a predefined set of rules. The first rule created was probably the addition where for any integer a, the integer (a + 1) is the least integer greater than a. This is based on observation of nature. It is therefore logic that it represents it well. From then on, it is still logic that all added subsequent rules will also fit with the physical reality.

raphalbatros
Logic Cloud
The physics-mathematics relation has been discussed quite extensively in the philosophy of science using a number of different approaches and from a number of different perspectives. In one approach, the applicability of mathematics to empirical phenomena is, very roughly, represented as a kind of structure-preserving map between two different structures. To my knowledge, however, there is no generally agreed upon account of this relation. I can provide some references on this subject matter if desired.

Gold Member
"In metaphysics, the problem of universals refers to the question of whether properties exist, and if so, what they are. Properties are qualities or relations that two or more entities have in common. The various kinds of properties, such as qualities and relations are referred to as universals." (Wikipedia, s.a.)

I would associate Wittgenstein here.
The question that comes up to me is how to categorize the mathematical or informatic theory of formal languages, which indeed has real applications nowadays. Isn't it an example how even metaphysics fit in areas of mathematical research? Not to speak about Gödel's results.
Most of it happened after Wittgenstein's lifetime. Nevertheless, to me these are valid arguments that the distinction between language and reality is artificial. At least it means one has to be extraordinary precise with the definition of meta-levels. (As to my knowledge philosophers still haven't come up with a satisfactory description of what is meant by reality. A term which probably each individual has to find an answer to by itself.)
@fresh_42 I don't know anything about formal languages. Can you explain or give a reference?

I would agree that what is meant by reality is not uniquely defined.

There seems to be an idea that there is this thing called the "real world" which is for unknown reasons well described by mathematical language. This to me is Nominalist thinking and according to Gillespie's book brings us back to the attitudes of the 14'th century.

My father, who was a doctrinaire Marxist, always spoke of the " material world " and this unseen eminence was responsible for the possibility of Science and Mathematics. I used to tell him that belief in a "material world" or if you like "real world" was essentially a profession of faith.

One thing that has always impressed me is how analysis of empirical situations leads to the definition of things that can never be measured or observed - e.g. a circle or a continuum of any kind -and also limits such as limits of Cauchy sequences. Mathematics seems to transcend measurement.

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andrewkirk, write4u and fresh_42
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2021 Award
@fresh_42 I don't know anything about formal languages. Can you explain or give a reference?

I was referring to this area of mathematics / computer science: https://en.wikipedia.org/wiki/Automata_theory or more mathematical https://www.iitg.ernet.in/dgoswami/Flat-Notes.pdf

There seems to be an idea that there is this thing called the "real world" which is for unknown reasons well described by mathematical language. This to me is Nominalist thinking and according to Gillespie's book brings us back to the attitudes of the 14'th century.
Well, I agree. We shouldn't fall behind Hilbert and Gödel, although the latter sort of destroyed Hilbert's program and idealism. But our common history is so rich on good philosophers that there is no need to add another point of view on reality. In the end it does not matter whether real numbers are really real, a lingual method, the way humans think or an algebraic object. As long as our bridges and buildings don't collapse ...

My father, who was a doctrinaire Marxist, always spoke of the " material world " and this unseen eminence was responsible for the possibility of Science and Mathematics. I used to tell him that belief in a "material world" or if you like "real world" was essentially a profession of faith.
Don't be too hard with your father. From his (philosophical) point of view he has certainly been right. There are simply some others with the same right of existence, i.e. a different view on what can be called reality. And Marx isn't the worst philosopher. Perhaps not quite from a mathematical standpoint, but overall his thesis still hold.

One thing that has always impressed me is how analysis of empirical situations leads the the definition of things that can never be measured or observed - e.g. a circle or a continuum of any kind -and also limits such as limits of Cauchy sequences. Mathematics seems to transcend measurement.
Yep. Most of us are Platonists ... (claimed by V. Strassen)
An interesting statement, by the way. Measurements or more general observations are the projection of reality which we can experience through our senses. Mathematics then is a transcendation filling the gaps with our mind, between molecules as well as on spacetime as a whole. And the best about your statement is the fact, that neither part depends on the term reality. Nice, I'll buy this.

write4u
@fresh_42 I don't know anything about formal languages. Can you explain or give a reference?
If I may, seems to me, in the physical sciences there is only one formal language, agreed to and used by all the sciences, Mathematics.There are some dialectic differences..
One thing that has always impressed me is how analysis of empirical situations leads to the definition of things that can never be measured or observed - e.g. a circle or a continuum of any kind -and also limits such as limits of Cauchy sequences. Mathematics seems to transcend measurement.
Yep, it's the way things work and our excellence lies in the cognition and understanding of these fundamental forces to an extend we can make them work for us. (landing on the moon).

Nature's functioning language is not in grunts and clicks as in human language, it is expressed as an interaction and results of measurable values and sets of values. As one scientist said, "if you ask the correct question and ask it nicely, you will get a correct answer"

IMO both Time and Mathematics are transcendental properties of the essence of the universe itself., IOW. Universal Potential becoming expressed in a naturally ordering environment is the most efficient (perhaps the only) way to reach the state of expressed probabilities of human exploration of the Universe..
As Tegmak says the simplicity of these mathematical functions which both affect physical things and IMO, creates measurable time in the process.

raphalbatros
"Mathematics can never prove anything."
Strange. That is mainly what I think about physics.
Well, by definition, from mathematics, you can only deduce things. Any deduction is based on axioms, and axioms are statements or propositions that are established by humans. The only way a deduction could generate a truth is if the axioms it is based on are also true. It would require for humans to be able to establish true statements or proposition. But there is no such type of reasonning, which can provide true conclusions.
Physic is a science, and science, by definition, can only induce things. (If a phenomenon has respected these laws every billion times we observed it, we might as well conclude it will do it again the next time). Nothing can be proven by induction.

awholenumber
well , i was just trying to learn differential equation ...
i was preparing myself to ask a few questions about differential equations ...
its a bit silly , sometimes i like to make up themes for my studies ...
few funny things went through my head ...

https://i.imgsafe.org/7c08af2ea3.jpg [Broken]

i also think questions such as these can only be answered by people who knows differential equation very well ...

i was just trying to tell , that i like this thread ...

i also happened to read this ...

The Schrödinger equation is the fundamental equation of physics for describing quantum mechanical behavior. It is also often called the Schrödinger wave equation, and is a partial differential equation that describes how the wavefunction of a physical system evolves over time.

i was also wondering that if you learn enough differential equation , you might be able to understand things like " The Schrödinger equation " , the nature of reality ... and in the end how mathematics can describe reality so well ....

i am also looking for some advice on how to start learning differential equation properly ...??

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write4u
Well, by definition, from mathematics, you can only deduce things. Any deduction is based on axioms, and axioms are statements or propositions that are established by humans. The only way a deduction could generate a truth is if the axioms it is based on are also true. It would require for humans to be able to establish true statements or proposition. But there is no such type of reasonning, which can provide true conclusions.
Physic is a science, and science, by definition, can only induce things. (If a phenomenon has respected these laws every billion times we observed it, we might as well conclude it will do it again the next time). Nothing can be proven by induction.
That may be true, but IMO, the subject is not our use of universal mathematics, but the recognition of inherent mathematical functions (constants) of spacetime.