Mathematics is the language in which physical reality is described. This doesn't mean that any mathematical framework describes any physical situation-just that any physical situation is described in some mathematical framework.

So, yes, by construction, in fact, mathematics is not, just, an effective way, it is the way to describe the world.

We would call mathematics any language that describes the world.

I am not sure that the world could be described in a finite closed sets of sentences. I suspect that the full description would amount to creation.

We construct (some people say hallucinate) physical reality. It is a bad concept. Perceived reality is better. Here we come to the realm that is not mathematical, of course. Good, better, best, beautiful, ugly are personal concept. Science does not deal with personal concepts. But we are part of the world. So, science by its ambitions does not deal with everything.

The article and the Abbott paper on which is based are just an example of clickbait. To begin with, there is the famous sentence "the typical 'working mathematician' is a Platonist on weekdays and a formalist on Sundays" (which I first learnt from a paper by R. Hersh worth reading: https://ac.els-cdn.com/0001870879900185/1-s2.0-0001870879900185-main.pdf?_tid=ba1c57bf-e900-4a54-a158-7a0f6a5f90de&acdnat=1526357215_2b68abde5288f2501168bf4cf1818f50), so no, not every mathematician thinks that mathematics exist outside the human mind. Second, the definition "Maths is effective when it delivers simple, compact expressions that we can apply with regularity to many situations" seems to be a joke. Maths is effective when it solves a problem and allows us to control the outout of a physical process, something that Abbott should know well given his training in Engineering. Probably all the Maths an engineer sees are as ugly and non-compact as they can be, but they are really effective and useful by any other human standard. Just look at the screen on your computer and think about the Maths behind it, from the Bezier curves used to display those wonderful Type-1 and OpenType fonts (those who have known the ancient bitmap fonts of the 90's should know what I am talking about) to the solution of the transmission losses equation needed to maintain a steady current around 125V (or 220V, if you are in Europe). Try to do all of this without Maths. I do not think that the mathematical description of physical reality has any other shortcoming aside from the difficulty to solve the problems it poses.

The main shortcoming is that mathematics doesn't describe physical reality, it describes physical theories. Reality can be understood only through theories, and that corresponds to our cognitive limitations.

Dear Claude Pierre,

''Reality can be understood only through theories,''

Most of what we understand is not through theories. The toddler learn to walk, understand how to walk, not through theories. Following the same reasoning, I would say that 99.99% of what most people know is not through theories. I would even say that altough natural scientist do understand through theories but it is not only through theories. Through their experience they also developed an instinctive understanding , tacit knowledge which guide them but which they would not be able to articulate explicitly in terms of theoretical knowledge. Even mathetician may instinctively feel a connection between two field of mathematic without at first able to be able to explicitly describe that connection. Our body is the product of hundred of million of living experience on this planet and most of that knowledge about reality is the spring of all our intuitions in whatever human activities, including the scientific ones.

LIke any science, any language, mathematics has limitations. Some limitations are intrinsic to what it is and can't be overcome and some limitations can be overcome by developing it further. Our natural languages make references to our senses and kind of human experiences and so are subjective, relying on our bodily experience. Mathematics fundational limitation is that it makes no such references. What it express is self-contain into its realm of definition. Existence is equivalent to definition. All references to reality, connection to our body, have been removed. This make it the most limited of all our language but the only one which we control totally although some cracks totally out of our control might be conceiled.

Mathematics, per se, describes nothing but itself within its own domain. It is the application of mathematics by human intelligence and reasoning that produces a description of what we perceive to be physical reality. The shortcomings are not within mathematics but within our ability, or inability, to achieve understanding of that which is the world about us. To the extent that we can claim understanding we have invented the mathematics as needed to undergird that understanding. The need for a mathematics to support the theory of general relativity culminated in the full development of tensor calculus as did quantum mechanics in developing the mathematics of linear vector spaces. The shortcomings, if they be, are within us.

“But it is not at all true that in pure mathematics the mind deals only with its own creations and imaginations. The concepts of number and form have not been derived from any source other than the world of reality. The ten fingers on which men learnt to count, that is, carry out the first arithmetical operation, may be anything else, but they are certainly not a free creation of the mind. Counting requires not only objects that can be counted, but also the ability to exclude all properties of the objects considered other than their number – and this ability is the product of a long historical evolution based on experience. Like the idea of number, so the idea of form is derived exclusively from the external world, and does not arise in the mind as a product of pure thought.

… But in order to make it possible to investigate these forms and relations in their pure state, it is necessary to abstract them entirely from their content, to put the content aside as irrelevant; hence we get the point without dimensions, lines without breadth, and thickness, a and b and x and y, constants and variables; and only at the very end of all these do we reach for the first time the free creations and imaginations of the mind, that is to say imaginary magnitudes. Even the apparent derivation of mathematical magnitudes from each other does not prove their a priori origin, but only their rational interconnection.

… Like all other sciences, mathematics arose out of the need of man; from measurement of land and of the content of vessels, from computation of time & mechanics. But, as in every department of thought, at a certain stage of development the laws abstracted from the real world become divorced from the real world and are set over against it as something independent, as laws coming from outside to which the world has to conform. This took place in society and in the state, and in this way, and not otherwise, pure mathematics is subsequently applied to the world, although it is borrowed from this same world and only represents one section of its forms of interconnection – and it is only just precisely because of this that it can be applied at all.” F. Engels, Anti Dühring, International Publishers N.Y., p. 45-46 (1939).

Dear Abdul,

I disagree with the statement: "But it is not at all true that in pure mathematics the mind deals only with its own creations and imaginations." We must first define "pure mathematics" and distinguish it from its primordial discovery or invention. "Number" arose out of experience with the real world but "number theory" in "pure mathematics' went way beyond any numbers that we commonly use in our daily lives. So my "mathematics, per se" is "pure mathematics" regarded as an abstract system of internally logically consistent deducible statements that may or may not have revelance to nature. "Pure mathematics" is concerned only with itself however we certainly can extract from or develope within "pure mathematics" those aspects that have application to our perceived physical reality. That is, there is "pure mathematics" and "applied mathematics" and the second is a subset of the first.

Dear Dwight,

The quote of Engels speaks for itself! I just posted the materialist dialectical view of the origin of mathematics as the creation of man; in support of the contention of Dr. Abbot. For more than a decade I have written against mathematical idealism in modern theoretical physics - in books, journal articles, public forums etc from a materialist dialectical perspective. Specially, my booklet, “The Einsteinian Universe? “ (now in its Second Ed.), contains a chapter, “The Magic of Mathematics”. https://www.amazon.de/Einsteinian-Universe-Dialectical-Perspective-Theoretical/dp/B01HCASNJK

Others including the mathematician Reuben Hersh has now spoken out against the Platonic view of mathematics in his book, “What Is Mathematics,Rreally?” Where Reuben Hersh argues the contrary, “that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. “

https://www.amazon.ca/What-Mathematics-Really-Reuben-Hersh/dp/0195130871

Astrophysicists Ellis and Silk have raised alarm about mathematical idealism in a letter to Nature, “Scientific Method: Defend the Integrity of Physics”. http://www.nature.com/news/scientific-method-defend-the-integrity-of-physics-1.16535

You claim, “So my "mathematics, per se" is "pure mathematics" regarded as an abstract system of internally logically consistent deducible statements that may or may not have revelance to nature.”

Of course any thought is abstraction and “pure mathematics” itself is abstraction from something, however obscure that something is, it does not have an a priori existence!; Even without internal consistency, wild imaginations of mythology have origin in the material world! You cannot imagine anything that is not induced (even if very remotely) by a real entity. You can of course in imagination join together various isolated parts of reality and make something “new” that does not exist - a unicorn, a horse with wings, a human figure with a the head of an elephant and so on or even multidimensional space etc.; but did anything arise from purely absolute thought that did not relate to any physical stimulus to the human brain – a material entity?

The Platonic view (as well as that of the mathematical idealism of modern theoretical physics led by Albert Einstein) posits that mathematics has "a priori" validity or existence and is the determinant of the universe. According to the Platonic view, mathematical forms and notions exist somewhere in the unknown realm; only a receptive and highly tuned mind can reach that realm to appreciate the truth and the beauty of mathematics. Modern theoretical physicists have adopted the Platonic view of mathematics and are aspiring to find a “theory of everything” of the universe– a mathematical formula that can be written on the surface of a coffee mug! We have titles like, “Our Mathematical Universe” by top grade physicists. The slogan “Truth is Beauty and Beauty is Truth; “Matter is a Myth” , because mathematical ideas are the only reality! Compared to mathematics, anything else, like philosophy is just trash. “Philosophy is as useful to physics as ornithology is to the birds” or “The Unreasonable Ineffectiveness of Philosophy in Natural Science”: etc., etc. Do you agree with all these!

Hello all,

The imagination of the Universe is way bigger than that of humans, so mathematics will bring less surprises than physics. Surprise is information, so physics is more informative.

To answer the question, on what are the shortcomings of mathematics to describe physical reality, mathematics is not bound to reality, while physics is. So, mathematics is a ping-pong table and physics is a ship -- they are of different types, it does not make sense to compare them. Having said that, physics and mathematics need each other. A ship can carry lots of ping-pong tables.

Cheers, Ed Gerck

First of we need to define the term 'physical reality'. Physical reality might encompasses several concepts and the most common concept is energy. Study into the concept of energy is known as physics. Physics subdivided into general physics and modern physics. Mathematics is the language used to study physics.

The shortcomings of mathematics to describe physical reality can only be placed at the physicist using and interpreting physics with certain math... that in (too) many cases has nothing with reality in common... best example is general relativity...

I don't know you, but I learned to walk through theory. I had many more theories about walking, that a toddler naturally master very well. Only minds perverted by mathematics lose this innate ability.

Dear David,

"What are the shortcomings of mathematics to describe physical reality?"

There are no shortcomings in mathematics.

The shortcomings are the wrong Natural philosophy that is used to describe Nature in the best possible way.

The mathematics are just the consequence of putting the ideas in a rigorous way, and to try deduce predictions from the ideas.

There are numerous examples of the wrong Natural philosophy that is used in nowadays kinematics, gravity, astrophysics, cosmology, and the impact is even sensible in electromagnetism.

With best regards,

Thierry De Mees

@Brassard: "The toddler learn to walk, understand how to walk, not through theories."

Learning to do something is not the same as understanding. If you require consciousness in understanding, that is.

If you don't require consciousness then learning to do something is also applying a theory to the world. The lion hunts on the theory he will succeed, even though he does not know that. Evolution is the continual development of theories by nature predicting that the newly developed evolutionary feature will help its proprietor to survive better. If the theory is wrong, the feature is eliminated, because the organism does not survive, if the theory is not wrong, the feature helps survive. In that sense, evolution represents intelligence without consciousness.

So if you do require consciousness in your definition of understanding then understanding is through theories and there is a lot of learning without understanding. On the other hand, if you consider any learning understanding, be it conscious or not, then you should accept that there is also theorizing without consciousness. And then the proposition that all understanding is via theories may not be wrong, after all.

There is some really deep philosophical thinking in the above responses. I have a simpler way to look at it. Logic seems to be inborn. If A implies B and B implies C then A implies C. We get that without any physics or engineering. It seems to be an inborn concept. Maybe there is some deep philosophical explanation of how that became inborn, but I just accept that it is without trying to explain its origin. Math extends this inborn logic by using language that simplifies the concepts, and by recording (by listing theorems) what we already figured out so that these things that we already figured out become tools for figuring more things out.

I am not really sure that the question is meaningful. If we take the position that the question is really "What are the shortcomings of mathematics as a way of understanding reality as we perceive it?" then the limitation of our perceptiveness is obvious. Of course we mustn't forget that there are (and must be) true things that are not provable ... Gödel's Theorem is obviously going to admit "real" things that are not accessible from the axioms. Whether we can perceive them is another matter.

Perhaps a better question might be "How useful is mathematics as a tool for exploring the world we perceive" , and the answer is likely to be "better than any of the alternatives that present themselves."

"What are the shortcomings of mathematics to describe physical reality?"

Mathematics can only deal with quantity not the quality of the objective reality. But it is only quality that can represent the enormous variety and the phenomenology of the universe! Quantity in a limited way is useful for the dynamics of the universe where quantitative change leads to qualitative leap in the evolutionary process of everything in the universe.

Mathematics is miserably limited even in the realm of quantity, because it can deal only with very simple and gross systems of everyday life and classical mechanics to Newtonian mechanics by extending iteslf a little more using statistical methods. It fails totally even in a bit more more complex terrestrial phenomena like the biological systems, for example.

Beyond everyday life experience, i.e., in the microcosm and macrocosm, mathematics has to abandon the material realm to transcend to the realm of thought (to gain some qualitative attribute), through the unlimited extension of idealized mathematics; where the materialist basis of objective reality has to be replaced with abstract Einsteinian “continuous fields” (spacetime, Higgs, quantum etc.) - matter becomes myth, mathematics turns into mystery or magic and science turns into theology!

This proves the dialectical law that any truth (mathematical truths included) when extended beyond certain limit, either turns into its opposite (myth) or becomes an absurdity. The worshipers of idealized mathematics especially in modern theoretical physics set no limit to their trades and can very easily roam around from the microcosm to the macrocosm riding on the magic carpet of mathematics just to satisfy their fantasies and become great “scientists” in the bargain!

Mathematics is the language for the expression of relations. Quantities are only a tiny aspects of this relational world expressed in this language. Whatever is not a relation can't be expressed in this language. There is another stange thing about the mathematical language is that it is also said to be a world. Take English. Nobody would say that English is a world. It is used to talk about the world. When we make English sentences describing aspect of the world, it is generally not an objective construction self-contain into an English world. Only human beings can make sense of such sentences because it uses words whose meaning has to be refered to an actual human experience. Mathematics is structured to be self-containt and nothing in it is refering to anything outside of it. So it appears to be the case. It is very unlikely the case given that we can model reality using this language. If the relations within mathematics were not already existing within the world, how likely such a language would be effective in modeling relational aspects of reality? I think the answer is: very unlikely. So, although mathematics is expressed as if it is totally unrelated to the world, it is on a superficial level, self-contained, all of it came originally from our world but all of this is conceiled in the mode of expression of this language. If it is the case then mathematics is a theory of the world that pretend not to be a theory of the world, that pretend to be outside of the world and still able to express our world. It is the original sin of platonism.

"Understanding" is fundamentally a synonym of theory. We understand something by splitting it in elements that we already know, or that we can understand. Why does the water freeze? Because it is made of atoms that move with respect to one another, and that can stop moving if the temperature is low enough. Here we have used the elements atom (in the original meaning,) motion, and heat. These elements are part of theories, namely atomism, classical mechanics, and statistical thermodynamics. In the physical meaning, there is no other way to understand. There is no answer to the question why?, only to the question how? When understanding how to walk, the elements are not that fundamental, but they can be split in turn in more fundamental elements, neurological, physiological, chemical, physical etc. That's how science began, with very crude concepts, that yet constitute theories. Today in physics, only theories formulated in the mathematical language are considered as such, but all that is required is that they be logically consistent and precise.

So mathematics is but a way of expressing theories, making sure they be consistent and precise. Reality is not of mathematical nature, because it is not a language: it is not the signifying, it is the signified. Per se, mathematics brings nothing more than consistency and precision. So, what describes reality is theories. The shortcomings of mathematics are the same as the ones of theory. Theories are limited by our cognitive abilities, while mathematics is not limited in describing theories. That's what is mixed up with "unreasonable effectiveness."

Louis Brassard,

English can be used to write a poem, and this poem doesn't need to refer to something in the world. Its own structure: rhymes, rhythm, sonority is self sufficient. Similarly, mathematics has something of poetry, and nowadays, it is often used to dream physical reality instead of describing it.

If we come to the original question and the paper that was appended there (https://phys.org/news/2013-09-mathematics-effective-world.html ), to my understanding one of the main questions is if the mathematics has its own world of existence or not. And that many physicists in line with Galileo are quite sure that the nature is lead by mathematical laws or that physical reality is mathematically describable. In other words, there is a belief shared mostly among physicists that physical laws are in essence mathematical structures clamped down to our energy-matter world (so called real physical world). In this view it is not so important, if we are Platonists (a lot is written about this in the cited article), believing that mathematical laws live in their own world, detached from our well known experienced physical reality or Aristotelians, believing that somehow mathematical structures are embedded in physical laws, not having their own separate existence. The basic question is still, if it is possible ultimately to describe physical reality via mathematics. And as it has already been said in this line of comments, there are many aspects of physical reality that at least as yet cannot be mathematically adequately presented. Mathematics has a strong element of idealization and can only approximately cope with real physical processes in all their intricacies. And when we come to biology, for instance (still trying to describe and explain some special systems of physical reality, namely organisms), only a limited span of life’s aspects can be more or less adequately presented by mathematics.

But we should not forget that mathematics of today is not mathematics of tomorrow. New models will appear, maybe even radically new. Some of them may prove to cover phenomena that at present cannot be mathematically covered.

It is not obvious whether new mathematical system will appear in the future. Newton and Leibniz had to build their own mathematical tools for developing their theories. Modern physicists think that new mathematics will be necessary too, for instance Feynman for his paths integral, or in the domain of renormalization. But this hope of equalling the big physicists of the past will probably be disappointed. The reason is, mathematics didn't significantly changed between Euclid and, roughly, Newton, which represents a time span of 2000 years! Since, the logical and axiomatic paradigm triggered a considerable development in the 19th century, the golden age of mathematics, which we now call the modern mathematics. Many ways have been explored because there wasn't the constraint of corresponding to something real, only logical soundness was required. The progresses have been so important that a unification has been tackled with the categories. Consequently, in all likelihood there is nothing left for the physicists to discover something new, they should use what is existing, and that is already a formidable task.

Math extends our inborn logic by providing a language that adds clarity to concepts, and by keeping a record of theorems already proven that are now tools for proving future theorems. But math is still limited by the human mind. If there are aspects of physical reality that are beyond what the human mind can understand (insert your own opinion here), then math has limitations regardless of how well educated we become and regardless of how many theorems we prove. I agree with Randall Gray's earlier answer in that math is probably the best option that we have so far.

Claude,

''English can be used to write a poem, and this poem doesn't need to refer to something in the world. Its own structure: rhymes, rhythm, sonority is self sufficient. ''

Poem are typically not realist but the words used in poem refers to the world but poetically so, in a way that renonate to our senses and emotions, this is even closest to what is most important for us in the world. I have no doubt that deep down, mathematics also refer to certain aspects of our senses , otherwise we would have no way to imagine and understanding it.

Igor,

''The basic question is still, if it is possible ultimately to describe physical reality via mathematics.''

I say NO. Mathematically expressed physical laws are not description of physical reality. They are approximate descriptions of possible ways to interact with limited validity. Physics provides us with mathematically expressed interface of interaction. What is beyound these interfaces, reality itself , is totally beyound any of our descriptions.

@ Louis Brassard

You have made some really interesting and insightful comments/answers to my question. Thank you!

Mathematics is a great tool and a great discipline to study our grand universe which reveals itself sometimes mathematically via our current physical laws.

"Sometimes less is more, and sometimes more is less."

With kind regards, Dave.

Hello all,

Mathematics is a poor way to describe physical reality, but it is better than any other existing alternative.

Cheers, Ed Gerck

I define the physical universe to be that portion of reality that can be perceived, either directly or indirectly, by at least one of our external perceptions (vision, hearing, etc.). "Indirect" includes the use of laboratory instruments that can detect things that we cannot see, but we still have to perceive the instrument reading to reach a conclusion (some experts in quantum mechanics might say that reality has not been defined until a consciousness has seen the result). "Indirect" also includes any cause-effect relationship in which the effect can be perceived by one or more of our external perceptions, and logically (based on our understanding of physics) implies the cause. The physical universe, as I see it, does not include all of reality because there are other kinds of perceptions (internal perceptions, like the awareness of our own thoughts), and maybe there could be still more external perceptions if humans become more evolved.

But the physical universe sure does cooperate with math and logic. If A implies B and B implies C then A implies C. This could not happen without support from the physical universe. A lot of the physical universe can be predicted from math, together with postulates (albeit, always changing) of the physical laws. I agree that math conceived by the human mind has limitations, but it is the best thing that we have so far for understanding physical reality.

The methods and abilities of mathematics improve each day. With the help of mathematics, we can describe increasingly complex processes and phenomena. This is precisely the reason for progress in engineering and technology. The process of perfection is endless, as the possibilities of mathematics are endless.

The most prevalent shortcoming is the assumptions that accompany the mathematical models describing physical realities. These assumptions hinder the predictability of physical reality through mathematical formulations.

Mathematics is the language of nature. Mathematical models can describe physical models but due to simplifying assumptions used in developing the models, the match between mathematical and real earth models is not always perfect but differs slightly depending on the degree of complexity of the earth model.

Why should there be shortcomings of mathematics to describe physical reality?

1. Physics you do first, without mathematics, i.e. you try to describe qualitatively the phenomenon.

2. Then, you try to fit a mathematical model, taking into account the factors you think that influence the model.

3. Then you calculate quantitative relations and see whether your model predictions are confirmed by the experiment.

4. Finally, if everything went O.K., you see what OTHER predictions does the model, and if they are confirmed by the experiment.

At any step, if the experiment does not confirm the model, you have to modify the model. So, why should there be shortcomings?

The formerly described methodology roughly covers the world of physics. It is weak in covering biological phenomena, because of high non-linearity of the majority of biological phenomena, spanning simultaneously on many levels. Yet, organisms are still a part of physical reality. Are there emergent laws? How to formulate them? How to cope with them? How to delineate their ontological status? Do they appear from pure potentiality, when a certain system comes to being? There are still many questions that belong to the possibility of ultimate mathematical description of the physical reality. Not to mention phenomena of consciousness that in the last analysis also belong to the physical world.

Igor,

You said that the methodology I described - in fact is not methodology, it is the spiral of the research - roughly covers the world of physics. You motivate the criticism invoking non-linearity. I don't see any contradiction between the spiral I described and any linearity, or non-linearity, etc., of some equations. The spiral is the general way of how advances our research.

So, please, can you explain?

Then you ask

"Yet, organisms are still a part of physical reality. Are there emergent laws? How to formulate them? How to cope with them? How to delineate their ontological status? Do they appear from pure potentiality, when a certain system comes to being?"

If mathematics would answer these questions of yours, you wouldn't need experiments anymore. "Are there laws?" you ask. First of all, organisms are described from two points of view: anatomy and physiology. For getting both of these descriptions you do hypotheses, and then experiments. The mathematics come after that, when you try to formulate a model. The mathematics casts your model into precise quantitative relations.

As to "phenomena of consciousness", don't forget that the human body is a chemistry lab. All the phenomena in it are ultimately based on physiological processes, nothing else. So, I refer you to the previous paragraph in my post.

The role of the mathematics is to express quantitatively the relations between the items in your models. But, for finding what are these elements you have to make hypotheses and test them by experiments. The mathematics won't replace the hypotheses making and the experiment. However, when testing the formulas, you may find, eventually, that something doesn't fit, s.t. you would have to make additional hypotheses, etc. As I described in the spiral.

Aleš Kralj,

Bravo! An answer simple, clear, and correct!

The shortcoming of mathematics to describe the physical reality is that it is based on the symbolic logic that our minds have found useful to understand the physical world at our level. We have seen the limitations of symbolic logic at the Quantum and Cosmic levels. We also see this in the Einstein's Special Theory on Relativity. thanks.

Dear Prof. Cole and other Readers,

I would like to take this opportunity to popularise our solution of the Millenium Mathematics Problems by our low dimensional Physical discovery of AGGNNetworks using String Theory. This is a Mathematical Sciences research to solve a perplexing set of mathematical problems to solve String Matching Field Theory properties while at the same time using the application of the Field Theory to kind of experimentally verify the proofs. This is a String Duality Method of proving theorems while at the same time forming the experimental system in computer science to solve them. Puritan Mathematicians may not like this answer but it does show that discontinuity in logic can be solved by mathematics too in the method of duality in String Theory. String Theory by definition thrives on discontinuity in fields and logic.You can take a look at our published proofs on (1) https://www.researchgate.net/publication/321299955_MILLENIUM_MATHEMATICS_PRIZE_FOR_THE_RAMANUJAN-HARDY-MALLICK-HAMBURGER-MALLICK_MATHEMATICAL_SCIENCES_SCHOOL_OF_THINKING_DIFFERENTIAL_STRING_THEORY ; (2) Results of our Research on the Millenium Prize problems

https://www.researchgate.net/publication/321051488_Results_of_our_Research_on_the_Millenium_Prize_problems

Soumitra K. Mallick

for Soumitra K. Mallick, Nick Hamburger & Sandipan Mallick

for the RHMHM School of Mathematical Sciences Thinking

USA, Japan & India (proposed)

Dear Alej,

It is rather the philosophic concept of the physical process that is studied, *translated* into a maths context...

Best regards,

Thierry De Mees

About 400 years ago Galileo wrote: “Mathematics is the alphabet in which God has written the Universe”. It was certainly a revolution because before the alphabet of the Universe was represented by philosophy.

Now nevertheless it is manifest that it doesn’t regard any mathematical relation, that is it doesn’t involve any mathematical relation represents the physical reality. it happens when the mathematical relation is also a relation of physics. It doesn’t happen always. For instance the mathematical relation F=ma is certainly a relation of physics because it represents Newton’s law. The mathematical relation F=mv instead is correct from a mathematical viewpoint but it isn’t correct from a physical viewpoint.

Hence limits of mathematics, when it is correct from a mathematical viewpoint, are relative to the inappropriate use of mathematical relations in physics. This way the choice of appropriate mathematical models becomes fundamental for the description of the Universe besides the intrinsic correctness of the mathematical relation.



Dear Sofia!

I had no intention to criticize your approach. For sure it can give many valuable outcomes and insights. But from my biological experiences and occupation with possibilities of theoretical biology I know that organisms very quickly become too complicated systems to be solved with contemporary mathematics. You can do models of complicated system and processes; long ago at the Open University in Milton Keynes I witnessed such trials. I was told then that by a sufficiently complex system of differential equations and by an apt fitting of parameters, you can simulate anything. But is this enough, does it bring us closer to understanding of an actual biological process – that may change abruptly under different conditions, limiting our equation system, proving it works only under a limited sets of conditions, while we still do not know the essential mechanism behind. But sure, we can try laboriously to find the latter. But as it is most probably only a part of an unimaginably complicated web of all biochemical and physiological processes, taking place simultaneously on many levels, our new knowledge would still be very limited and my again prove to be inappropriate in a different set of physiological states of the studied organism.

As to the laws, I do not ask if there are laws, but if there are emergent laws, the laws that pertain only to a certain level of system’s organization.

wherever we see ordered behavior, not chaotic, there is possible mathematical formalization of the order, which is manifested by the system.

the actual complexity is not an obstacle, an example of this is thermodynamics, which goes from microscopic complexity to global generalizations in the thermodynamic limit.

another thing is that the" law"," order", according to which the living system behaves very different from the law of the inanimate system. And that's the difficulty - to understand this "living" order. What exactly does a live system do? What is its objective function? This question is equivalent to the question of what is the "meaning of life" in overall.

In the preceding comment I referred substantially to the physical universe in concordance with Galileo’s claim. I have observed discussion regards also the biophysical universe. Naturally the approach of biophysics is certainly scientific and if one accepts this approach then he cannot renounce mathematics in biophysics also in the order of an approximation that certainly is more reliable than any other approach. Science doesn’t search for the absolute truth but it searchs just for a truth in the order of a science in progress and of the scientific method that is used by human observers with their limits. The absolute truth instead is object of philosophy and of religion.

It is manifest that biophysical systems are more complex than physical systems and besides biophysical systems have metaphysical exigencies besides a physical structure. An apple falling from a tree doesn’t have metaphysical exigencies (at least it so seems) while for instance the “homo sapiens” certainly has metaphysical exigencies and aims besides a physical structure. The scientific study of biophysical systems can regard with reasonable level of reliability the physical structure of biophysical system while the scientific approach presents some problem with respect to the metaphysical structure of system.

From the physical viewpoint the most important aspect of a biophysical system seems to be the concept of “feedback” that instead isn’t present in a natural physical system. There are mathematical models for studying the feedback and they are used also in artificial physical systems. A system of equations can be able to describe the physical structure of a biophysical system but certainly no metaphysical structure. Naturally also for equations of biophysics, like for equations of physics, is valid a criterion of reliability that is connected with the order of approximation with wich the observer is able to interpret and to model the behaviour of physical and biophysical systems.

Mathematics has been called the language of the universe. Scientists and engineers often speak of the elegance of mathematics when describing physical reality.

Physics, and biology, are experimental sciences. Without experiment, there wouldn't be physics at all, that has nothing to do with correctness or existence of equations.

A biological system, or living being, doesn't obey the ergodic hypothesis of thermodynamics. The numerous processes all are dependent in a non trivial way. It can't be treated probabilistically, and in general no simplification is possible. In few words, a system of equation that would describe a living being is intractable. Therefore there is no mathematical biology. There is no way to claim that a reductionist view is correct, and there can be something else than plain mathematics and physics.

Now it is often claimed that science is "more correct" than, say, religion, because it rests on proofs. In the present case, there is no proof at all, and saying that any biological system can be described by a huge set of mathematical equations is not science, it is belief.

Mohamed Hassani

You don't answer to what I wrote, that shows you didn't understand me. Everybody here knows the scientific method, and precisely experiment is one of its links without which there wouldn't be physics at all.

The shortcomings of mathematics for doing physics is that it is purely deductive. That's why Poincaré didn't discover the relativity, while he had everything necessary for that: he followed a deductive approach. Einstein was looking for simple and elegant principles, that are however able to explain many facts (economy principle of Mach.) He thought that everything must have a reason (sufficient reason of Leibniz.) In particular, he was concerned with the difference in the description of electromagnetic effects according to the frame of reference. That is more philosophical, metaphysical, or especially inductive than mathematical. He could then deduce, rather than postulate, the Lorentz transformation and E=mc².

Dirac wrote that the future progresses in physics would be obtained through mathematics. In the same paper he proposed the magnetic monopole, that have though never been observed, falsifying his own prevision. Then ended his productive life as a physicists. Before, he predicted the positron apparently through mathematics, but his equation was but a direct consequence of relativity and of Schrödinger's quantum mechanics, that arose a short time before, and with heuristic arguments in what concerns Schrödinger.

In the post-modern era, beginning in the seventies, the advice of Dirac was predominantly followed, which gave such resounding flops like supersymmetry or grand unification theories that have been ruled out by experiment, and strings that have not even been ruled out. If mathematics are used in physical theories, that doesn't mean they are used to get these theories. That damages seriously the idea of consubstantiality of physics and mathematics.

Poincarè could not discover relativity because the relativity was discovered previously by Galileo.

Einstein didn’t discover relativity but he elaborated a different theory (Special Relativity) based on a different mathematical model of space-time: Lorentz’s Transformations. Also posclassical physicists (Fitzgerald, Lorentz, Poincarè, etc..) reached before Einstein the same mathematical model with different and incompatible justifications. This incorrigible conflict between postclassical physicists and postmodern physicists would have to inspire contemporaneous physicists to consider if effectively Lorentz’s Transformations are the correct mathematical model for describing space-time and questions of relativity.

Mathematics and experiments are two different stages of the scientific method and it is the guarantee of its validity that doesn’t have the claim of searching an absolute truth. Science and physics in particular aren’t in conflict with religion and philosophy that have different reasons and aims. Who is unsatisfied with answers given by mainstream physics has two roads: to search new solutions in the order of the scientific method or to search new answers in the order of philosophy or religion.

Poincarè could not discover relativity because the relativity was discovered previously by Galileo.

Einstein didn’t discover relativity but he elaborated a different theory (Special Relativity) based on a different mathematical model of space-time: Lorentz’s Transformations. Also posclassical physicists (Fitzgerald, Lorentz, Poincarè, etc..) reached before Einstein the same mathematical model with different and incompatible justifications. This incorrigible conflict between postclassical physicists and postmodern physicists would have to inspire contemporaneous physicists to consider if effectively Lorentz’s Transformations are the correct mathematical model for describing space-time and questions of relativity.

Mathematics and experiments are two different stages of the scientific method and it is the guarantee of its validity that doesn’t have the claim of searching an absolute truth. Science and physics in particular aren’t in conflict with religion and philosophy that have different reasons and aims. Who is unsatisfied with answers given by mainstream physics has two roads: to search new solutions in the order of the scientific method or to search new answers in the order of philosophy or religion.

Who is satisfied with answers given by mainstream physics has nothing to say.

Contrary to what is often claimed, Galileo didn't discovered Galilean relativity, it was… Poincaré.

Aleš Kralj

When science want to subdue religion, and so becomes an ideology, dark ages emerge too. There have been nazism, communism, and now there is capitalism. Science is about true and false, religion is about right and wrong, and art is about good and bad. It is amazing how so many people are unidimensional. When in the Eden, God said to the fist humans that they might eat the fruit of the tree of science, but not of the tree of good and evil. Thus the difference was proclaimed from the very beginning, but the humans don't want to hear, they prefer the slick words of the snake.

Aleš Kralj

My intention wasn't to despise anything or anyone. Please, respect people who don't think like you.

Aleš Kralj, you said “Religion is mostly about wrong. I can't recall any religion that is moral. Science is morally right as tries to tell the truth.« It is true that various peoples comminted cruelties in the name of God or this and that religion and that in many lands people of a different religion from the established one were (are) persecuted. It is in human nature and mostly it was not meant by the religion founders. In principle, religions tend to establish ethical relations between humans.

But there is another important issue here. Namely, science is not exempt to failures seen in the religious world. Even if science is envisioned as a bringer of light and trooth, many, especially established, scientists, tend to be rather dogmatic, self assertive and conservative. In other words, when the established paradigm is threatened, science may begin to behave as a fundamentalistic sect, persecuting the proponents of the threatening theory even on the level of the tabolid press. Just monitor various skeptics forums.

Aleš Kralj, you said “Religion is mostly about wrong. I can't recall any religion that is moral. Science is morally right as tries to tell the truth." It is true that various peoples comminted cruelties in the name of God or this and that religion and that in many lands people of a different religion from the established one were (are) persecuted. It is in human nature and mostly it was not meant by the religion founders. In principle, religions tend to establish ethical relations between humans.

But there is another important issue here. Namely, science is not exempt to failures seen in the religious world. Even if science is envisioned as a bringer of light and trooth, many, especially established, scientists, tend to be rather dogmatic, self assertive and conservative. In other words, when the established paradigm is threatened, science may begin to behave as a fundamentalistic sect, persecuting the proponents of the threatening theory even on the level of the tabolid press. Just monitor various skeptics forums.

Aleš Kralj, you said "Religion is mostly about wrong. I can't recall any religion that is moral. Science is morally right as tries to tell the truth." It is true that various peoples comminted cruelties in the name of God or this and that religion and that in many lands people of a different religion from the established one were (are) persecuted. It is in human nature and mostly it was not meant by the religion founders. In principle, religions tend to establish ethical relations between humans.

But there is another important issue here. Namely, science is not exempt to failures seen in the religious world. Even if science is envisioned as a bringer of light and trooth, many, especially established, scientists, tend to be rather dogmatic, self assertive and conservative. In other words, when the established paradigm is threatened, science may begin to behave as a fundamentalistic sect, persecuting the proponents of the threatening theory even on the level of the tabolid press. Just monitor various skeptics forums.

Aleš Kralj, you said "Religion is mostly about wrong. I can't recall any religion that is moral. Science is morally right as tries to tell the truth." It is true that various peoples commented cruelties in the name of God or this and that religion and that in many lands people of a different religion from the established one were (are) persecuted. It is in human nature and mostly it was not meant by the religion founders. In principle, religions tend to establish ethical relations between humans.

But there is another important issue here. Namely, science is not exempt to failures seen in the religious world. Even if science is envisioned as a bringer of light and truth, many, especially established, scientists, tend to be rather dogmatic, self-assertive and conservative. In other words, when the established paradigm is threatened, science may begin to behave as a fundamentalist sect, persecuting the proponents of the threatening theory even on the level of the tabloid press. Just monitor various skeptics’ forums.

Sorry, the Res.Gate reported an error, as if it could not upload my answer, but it apparently did many times.

Igor

Dear Igor,

You wrote : "... when the established paradigm is threatened, science may begin to behave as a fundamentalistic sect, persecuting the proponents of the threatening theory... "

Or, mutatis mutandis, the opposite : establishment scientists make it impossible to dissidents to offer any of their views in" respected" journals, whatever the evidence is that they give.

Best regards,

Thierry De Mees

Physical world tends to discontinuous on a small scale and continuous on a large scale. Math can handle it, but not in a unified function.

Aleš, in practice you are right, but what I wanted to stress is that in the scientific world we have a very similar behavior, which is expressed also in @Thierry De Mees comment. I know many examples where no quantity or quality of evidence suffices for a scientific doctrine to change accordingly. The history of science also abound with such cases. The principle of exclusion expressed in your comment does not hold only in the religious world, but unfortunately also in the scientific one.

All

The problem is how we relate mathematics to physics, and that we often overestimate mathematics in relation to physics:

1) Sagnac effect can be described as a) integration of speed along a line or b) as an integration of rotation over a area. They are mathematically identical, but only a) is valid in physics.

2) Faraday did a gigantic work regarding the ether and Maxwell only translated into mathematics, and got the majority of the honor.

3) Tycho Brahe did a gigantic work on planetary motion. Without him could expect to wait hundreds of years to find someone equal. Kepler just substituted circles by ellipses, which was almost self-evident, and got majority of honor.

With the best regards from _______________ John-Erik Persson

The mathematical model of space-time (Lorentz’s Transformations) elaborated by postclassical physicists (who support theories of ether) and by postmodern physicists (who support Special Relativity) proves this with reference to a simple physical situation:

If an observer moves inside a train with velocity u=5Km/h and the train moves with velocity v=100Km/h with respect to rails, the velocity of the observer with respect to rails isn’t v’=105Km/h but it is lower. Similarly if a ray of light moves inside of the train with the known velocity c, the velocity of the same ray of light with respect to rails isn’t c’=c+v but c’=c, i.e. in that case the velocity v of the train doesn’t exist.

These results derive from a correct mathematical use of that mathematical model (LT)of space-time. The question is if those results are correct from a physical viewpoint. The question seems banal but from more than one century it is cause of many discussions among physicists.

The new mathematical model, in place of the Galilean-Newtonian model of space-time, is justified, even if with different excuses, by postclassical physicists and by postmodern physicists as per two considerations:

1. the Michelson-Morley experiment doesn’t confirm the Galilean relativity

2. Maxwell’s equations are invariant under Lorentz’s Transformations.

I have good scientific reasons, and together with me many others, for claiming these considerations aren’t valid. An useful discussion would have to avoid dogmatic stances and would have to propose a neutral comparison. This clarification is necessary because in mainstream scientific environment this comparison doesn’t exist because Galilean-Newtonian transformations of space-time are considered wrong outgoing as per conclusions in points 1 and 2.

I hope an useful discussion is able to give further explanations about these simple physical situations and to allow to examine also more complex physical situations.

Hence in order to answer the question raised by David Cole, I think shortcomings aren't in mathematics when it is used correctly but in the wrong choice of the Mathematical model for representing the physical reality.

Dear Daniel,

The Galilea idea that c'=c has a physical reason. There is a regulating process that reduces c+v into c for waves. It is: ether.

Galilean relativity is only locally valid, because even if two distant galaxies moving relatively, will locally measure c, seen from either galaxy, the other's speed of light isn't c.

Michelson's experiment confirms it. Ether is locally bound with the Earth. It's speed is locally zero.

Best regards,

Thierry De Mees

Dear Thierry,

thanks for having remembered and specified the viewpoint of supporters of ether theories.

Daniele and Thierry

The Sagnac correction in the GPS system suggests, that a not rotating frame, with the same translation as the center of our planet, can represent the ether wind. However, there is another possibility, namely an ether wind that is spherically symmetric in relation to the center of our planet. This ether wind can also be united with the high precision in GPS, due to the spherical and concentric arrangement of GPS satellites and receivers. Such an ether wind can also explain gravity.

Best regards from ________________ John-Erik

Dear John-Erik,

The Sagnac effect contradicts the second postulate of Einstein’s Special Relativity and hence it is a further scientific proof of the invalidity of that theory, but the Sagnac effect doesn’t prove the existence of ether.

The Sagnac effect proves instead the insignificance of the mathematical model represented by Lorentz’s transformations. In fact Lorentz’s transformations, whether in the postclassical version of ether theory or in the postmodern version of Einstein’s Special Relativity, demonstrate the constancy of the velocity of light with respect to all reference frames. The Sagnac effect, like many other physical facts, denies this conclusion.

The Sagnac effect is explained correctly by an appropriate choice and by an appropriate use of reference frames that in the Sagnac effect are the rotating platform and the Earth.

Yes, Galilean transform is valid only locally, as Thierry said. Daniele talks about all reference frames. So, these frames are in reality a field. The ether wind is a field; and this field entrains bodies to move in such a way that the ether wind from distant bodies is compensated by acceleration in free fall. However, the effect from nearby bodies produces an ether wind in radial direction towards that body. Therefore, an artificial satellite reacts to gravity from Earth.

Regards _________ John-Erik

In actuality in my preceding comment I referred to all inertial reference frames, because it is known relativity is valid into the inertial field in which there aren’t acting forces among different inertial reference frames of the field. The inertial field hence is characterized by zero force.

Accelerated fields instead are characterized by different from zero acting forces. Gravitational fields are accelerated fields and consequently motion of bodies in free fall is caused by the force (Newtonian type) of the gravitational field that is compensated by the reaction force of the falling body.

In the event of natural planets, natural satellites and artificial satellites, orbital motion is caused by the compensation between gravitational forces of the central gravitational field and of the secondary gravitational field. I don’t see the necessity to make use of a unknown and undefined ether wind.

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What are the shortcomings of mathematics to describe physical reality?

In the light of various major or minor disputes, demonstrates also in commentaries to this question, via (correctly computed) mathematics we can validate either one thesis (let’s say of the ether wind) or its opposite (let’s say there is no ether at all). Mathematics therefore serves our ends and our perception of facts. It can bring us closer to understanding of the physical reality, but if our presumptions are wrong, it can lead us astray.

For many physicists, mathematics has become a new religion. No, the revealed truth is not in mathematics, that is wrong.

"What are the shortcomings of mathematics to describe physical reality?"

Rather the "strength" of mathematics is that its “shortcomings” are not realized or recognized by modern official physicists. Mathematics is a newly revived (since Einstein) alienation for mankind– a creation of man, but that has gone out of his control and like a Frankenstein Monster has come back to haunt and rule over physics. Mathematics is now considered the “language of physics”, as "the a priori determinant of the universe", the same way Platonic idealism used to view reality. Since the fantastic idea of the “Big Bang Creation” of the universe, this Frankenstein Monster has taken total control over modern theoretical physics (mostly Cosmology and Particle Physics): https://en.wikipedia.org/wiki/Our_Mathematical_Universe

The fact of the matter is that, mathematics is the epitome of the faulty world view of Causality. It is a very specific, highly limited and unidirectional, inherently mechanistic and deterministic tautology that strictly and faithfully obeys certain man-made axiomatic and self-evident truths. But very often those truths have no relation to objective reality. This epistemology does not look for new truth, but only wants to see (and prove!) the manifestation of the pet truths (it assumed in its premise in the first place) in the details of Nature. And un-surprisingly the Frankenstein Monster is flexible and efficient enough to find those assumed "truths" without fail. What is the best part of it all is that the physiciats can then readily do dedicated experiments to "prove" those truths !

Albert Einstein, in whose name the Frankenstein Monster rules over physics had this to say: “ In so far as theories of mathematics speak about reality, they are not certain, and in so far as they are certain, they do not speak about reality”. Geometrie and Erfahrung (1921) pp. 3-4 link.springer.com

Also this: “Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore". As quoted in the essay "To Albert Einstein's Seventieth Birthday" by Arnold Sommerfeld, "Albert Einstein : Philosopher-Scientist" (1949) edited by Paul A. Schilpp (p. 102).

And on geometry, the very foundation of modern cosmology: “Geometry sets out from certain conceptions such as "plane," "point," and "straight line," with which we are able to associate more or less definite ideas, and from certain simple propositions (axioms) which, in virtue of these ideas, we are inclined to accept as "true." Then, on the basis of a logical process, the justification of which we feel ourselves compelled to admit, all remaining propositions are shown to follow from those axioms, i.e. they are proven. A proposition is then correct ("true") when it has been derived in the recognized manner from the axioms. The question of "truth" of the individual geometrical propositions is thus reduced to one of the "truth" of the axioms. Now it has long been known that the last question is not only unanswerable by the methods of geometry, but that it is in itself entirely without meaning … The concept "true" does not tally with the assertions of pure geometry, because by the word "true" we are eventually in the habit of designating always the correspondence with a "real" object; geometry, however, is not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical connection of these ideas among themselves” Albert Einstein; Relativity: The Special and General Theory.. Methuen & Co. Ltd. 1952.

Abdul

Thanks for these good ideas. Yes, it is dangerous to invent (not discover) axiomatic pet theories that sooner or later get some kind of so called evidence:

a) I it is dangerous to define parallelism by a point that does not exist. Equidistance exists.

b) It is dangerous to define uniform motion by the fact that it can not be detected. Many failures does not prove impossibility.

c) It is dangerous to define light not to depend on the source. It does depend on the ether.

Euclid and Einstein told God what to do and started a search foe evidences and when they failed they turn that into success by stating that they have proved the negation instead. Negation is dangerous in definitions.

John-Erik

@ Christian Baumgarten "Well, "exclusion" is still different from stoning, burning witches or genocide. These things have been done in the name of religion (and also in the name of the pseudo-science of nazis and sovjiets). It is not so common in "normal" scientific discourse."

It is true scientists are not killed by some awesome authorities, but they may be severely humiliated, they may even be divested of their laboratory (cases of Widener F. Cope (committed suicide after having been bereft of his lab) in USA and Benveniste in France). Many scientists (I am speaking of good scientists not of some bad ones) that do not conform to the prevailing paradigm may be banished form the scientific community. It can be a very crude method of almost tribal-like exclusion without any judicial procedure (there is no court for science), no possibility to self-defend contra the establishment. It can resemble Galileo’s case.

Well, mathematics is effective, but with respect to what? We have nothing else allowing to compare the effectiveness of mathematics with. Worst, everything on which mathematics could be applied to see its effectiveness has been obtained and tested with mathematics, therefore it's circular reasoning. There is a hidden and manifestly false assumption that we know the real world, and we only need to see whether mathematics is able to describe it correctly. All this topic is utterly absurd, and besides irrelevant. We'll continue to use mathematics since we have nothing else, whether it is effective or not.

Nevertheless, there is one thing we are sure about, mathematics can't describe the Universe as a whole. Let us take the simplest example: the natural numbers. If we want to count all the particles of the Universe, we will not have enough fingers. Counting in mathematics is building bijections between sets. Now we have only one set of all particles, by definition there is no other Universe to build a bijection with. There are many paradoxes when we consider the scale of the Universe, one of which being precisely a fundamental paradox of mathematics: the set of all sets. There is also the entropy, the wave function of the Universe… That's why there is no way to decide, through reason, whether the Universe is infinite or not. Reason says it is infinite, observation says its size is 10^40. Mathematics is valid only locally, it isn't able to answer fundamental questions like the existence of God, finality, the anthropic principle… The Universe is certainly not mathematical.

Igor Jerman

We can't put aside pseudo-science because all science is epistemologically false. Of course, in hard sciences it is without consequences, but in medicine, psychology or sociology this could lead to dramatic issues. For some time is was thought that babies didn't experience physical pain, so that chirurgical operations were performed without anesthesia. There have been awful handling of persons that psychiatry declared insane. Further examples in natural sciences are easy to find. That happens when we claim to know, while most of the time it is but tentative false knowledge. We must keep in mind that science as proven and certain is about only comparatively nothing, science is mostly false. That's why the precaution principle is more and more implemented.

As for religion, its harmful effects have been greatly exaggerated by contemporary ideologists. There have never been religious wars, they were always conquest wars for territories or natural riches. In Israël it isn't a religious conflict, only that the zionists try and make it believe as a mere propaganda at the service of their purpose. The Inquisition existed, but was not as gruesome as the protestants claim. In the whole, we see that most of the time, genuine religious conflicts occur in the same country, and between very similar creeds, like protestants and catholics in Europe, or sunnists and chiites in muslim countries. However, the stake is more domination and influence than purely doctrine.

“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” -- Albert Einstein.

Why would the great Einstein make such a quote? Does physical reality defy (or is suspect to) the laws (axioms, models, proofs, theorems, laws, etc.) of mathematics? Why?

On the other hand, the MIT Prof. Max Tegmark believes physical reality is a mathematical structure/object.

"Is Reality A Mathematical Structure? - Horizon: What Is Reality?"

Please make the following links below for details.

https://www.youtube.com/watch?v=PTF-hHGbQ6s

"THINK BIG!",

https://www.youtube.com/watch?v=_3UxvycpqYo

Relevant Project Link:

https://www.researchgate.net/project/Hilberts-Sixth-Problem-https-enwikipediaorg-wiki-Hilberts-sixth-problem

Mathematics have replaced Anschaulichkeit at the advent of quantum mechanics. But they could experience the same fate. Nature doesn't care of fig about our representations.