> I do not think it a coincidence that our formulations of physical laws are so

> naturally expressed mathematically, I think it means we need to consider the

> possibility that the laws of physics themselves are an extension of

> mathematics.

I'm not sure I agree with that. As a colleague of mine noted recently, while physical laws are continually being adjusted, tweaked, expanded, or sometimes even overturned, properly proven mathematical laws (theorems) have never had this occur (certainly, some conjectures have been disproven, but no properly accepted and rigorously proven theorem has as far as my mathematician colleague is aware). He added that, to him, pure mathematics was as close to the Platonic ideal as one could get.

But let me give you an actual example as well that came up during a conference that just ended. It's a rough description, but it will suffice for what I'm trying to say.

An open question in quantum information theory relates to something known as quantum channels. Pretty generally, these are communication channels and could be just about anything. But some of them have non-unitary behavior meaning we can't approximate them using a bunch of unitary operators. It has been conjectured that if we take more and more copies of one of these channels, we might be able to get closer to an actual unitary representation. Specifically, in the asymptotic limit, the conjecture assumes it is possible.

Now let's put on our experiment's hat. As useful as this conjecture sounds, it means to get a perfect unitary representation we'd need an infinite number of copies of this channel. But this makes no physical sense. No experimenter can ever physically achieve this. Nevertheless, the conjecture exists and may end up being proven.

Another way to put it is that we have plenty of mathematical statements and proofs for situations that are completely unphysical.

Now consider that many - if not most - physical laws can be modeled in multiple ways mathematically, i.e. it is possible to model some physical laws using two seemingly unrelated mathematical procedures. Which procedure would the physical law be extending? In non-relativistic classical theories, there are two types of physical laws: laws of coexistence and laws of succession. In quantum mechanics these become selection and superselection rules. But this puts a serious limitation on mathematics if it is an extension of it since, for example, the former would imply that non-relativistic classical theories are only describable by equations and inequalities and yet there are so many other mathematical structures that can be used to describe even the simplest things in nature, not to mention the fact that mathematics itself has further generalizations for equations and inequalities that work in classical situations (groups, categories, etc.).

I hope that made sense. It's past my bed time.

a month later

Some random thoughts on this issue of the place of mathematics in physical realism.

IMO, a rudimentary system of numbers and basic arithmetic to relate these numbers to one another were formed as a tool that helped us with our survival. Those who could make sense of a number system were in a better position to plan and make predictions which were of benefit to individuals and communities.

I have a bunch of bananas lying around. If I eat a banana there will be less bananas in the bunch. How can I always keep track of how many bananas I have in total without having to constantly go back to the basket and look at them?

Someone wants to trade me coconuts for bananas and will give me less coconuts than I gave them bananas. Since I have a lot of bananas I need to be able to have a systematic way of dealing with such questions as, is it worth it for me to make this unbalanced trade?

A basic abstract number system and a system of counting is the result of trying to be able to make practical use of keeping track of a collection of objects like bananas. You end up with the very generalized and abstract notion of the concept of a number.

In my counting system, I know I have 12 bananas and will not be able to get anymore until I go on another hunt for bananas many moons from now. How many can I eat every day and make sure I have enough left to keep me going until that time? I will then need to abstract the concept of number even further and relate numbers to the occurrences and regularities of natural phenomenon. The Sun regularly returns to a fixed point in the sky in regular intervals and this is how we plan our days and how we know when it's time to go in search of bananas. The period when the sun appears on the horizon, disappears, and appears again I will call One day just like One banana represents what I have in my hand when I hold a banana. In three suns, the sun will have appeared three times.

I have 12 bananas but it will take 24 suns before I can go search for more bananas. This is when the notions of class distinction and category start appearing. One needs to go outside the bounds of the concept of integers and the real number system starts to gradually form. You understand that you will need to only eat parts of a banana each day. How many parts can I eat each day? I can slice each banana into two parts. I cannot call it two bananas because it is really just a banana cut into two segments. It is now less than one whole banana but more than nothing. I have no number in my counting system that represents this concept or the concept of nothing. The real number system then starts taking shape.

Although there are two objects which resulted from my slicing the banana down the middle, it represents two equal parts of a banana, not two bananas. I will have one-half of a banana and represent this idea not by 1 or 2 but by ½ -- this odd-looking number represents one of the two separate but equal parts of a banana. ½ day represents the sun going from horizion to horizon.

Wow. After a while, given my observation of the order and regularity of the things in the sky, I can start to use my understanding of this new system of numbers to predict durations of astronomical events and relate one to another. A very rudimentary form of empirical science then starts taking shape. After a while, I can start to predict things that happen in the real world using only these numbers and the relations between them.

It is no coincidence that numerology flowed out of Astrology and this is why so many figures in history have attributed mystical properties to numbers. The uncanny ability to use the number system to predict events in the heavens was seen as indicative of divinity. The numbers themselves became associated with the heavenly realms that existed in the firmament.

As abstraction becomes more complex, there is a systematic need to formalize such studies so everyone is working on the same page. Those concerned with the subject start talking about systems of numbers, sets, objects, theorems, axioms, proofs, etc. Eventually, you unite numerical and geometric concepts with intuitive pictures of space and distance and form more general and abstract systems like Cartesian coordinate system. Further systems of abstraction are introduced and you eventually get to the formal structures we see today.

Nature exhibits order and structure. If it were not so, we would not exist to talk about such things as whether mathematics exists as an ontological part of the natural world. Why nature exists order and structure instead of simply being a sea of random chaos and why it exhibits the particular structure it does is another question for another day. Anything that is ordered and structured can be mapped and analyzed. Mathematics is the toll we have developed to do just that. Mathematics is not something which exists as part of nature. It describes order, it does not create it

The number One or Maxwell's equations do not exist in nature anymore than the words 'Apple Pie' exists in nature. Someone may say, 'You're wrong ! The number one does indeed exist' I can then simply say that one is just two halves of a whole. It is the two halves that exist in nature, not the whole we associate with the number One. It is the number representing the concept of one- half that is fundamental. I am not looking at one banana, I see two halves which comprise one banana. Anyone can arbitrarily try to define what numerical or mathematical entity it would be that exists as part of reality in associating this entity with any member of a real-world collection of objects. Numbers and Mathematics are abstractions.

  • [deleted]

William,

I am glad to have stumbled upon this most interesting and well-formed question.

Just recently, I had an exchange with Ian Durham over the nature of mathematics in which we agree that mathematics is a language. If language is independent of meaning, however (and I think it is) then the mapping of linguistic symbols to physical phenomena is a process of evolution, as you suggest, such that "certain statements become true as they become operationally possible."

"Operationally possible," however, necessarily assumes an agent operator, whether some self-sustaining mechanical computer, or a humanoid being. In either case, this agent has to be capable of both inventing theorems and writing proofs. A state machine (Turing machine) is not so endowed; the changing states of natural things may be universally computable, therefore, without being operationally possible. In other words, some agent might be able to exhaustively describe every state that ever existed or ever will, without being able to prove the unity of these states. Mathematicians already know this limitation -- e.g., Euclidean geometry remains true in the flat plane, while non-Euclidean geometries incorporate different, and in many ways richer, domains. That doesn't render Euclidean geometry untrue even in the physical sense; physically, e.g., we have general relativity which describes reality as mostly flat spacetime at the local scale and curved in extremis in proportion to the presence of mass. The geometries aren't compatible, but the theory incorporates each in its own domain -- and the theory, being a scientific theory rather than a mathematical one, cannot be proved (i.e., cannot claim a completely closed judgement of truth, as mathematical theorems do).

In such a context, I also can't imagine what "supra-natural" mathematics could be like, because the operator agent occupies the same domain limitation as the results that (he or she, or it) describes -- the operator is a three dimensional creature with a four dimensional brain-mind. The language is finite but unbounded.

So far as "computationally possible" goes, we have results that bring serious questions to the table concerning the reliability of digital computing. Gregory Chaitin's constant (Omega) which describes the halting probability of a Turing machine, is an uncomputable number whose value is dependent on the program language running the algorithm. If even arithmetic is to some degree uncertain, then well ... your question about the status of arithmetic at the Planck density may already be answered.

There are many reasons that I reject the Platonic ideal of mathematics, but I think my main objection is same one that Deists have to the existence of a personal god. The evidence favors a self-organized world, and language -- even mathematical language -- does not therefore live in a world apart from the meaning that the world communicates. The meaning belongs to nature, but the symbols and vocabulary are ours.

Tom

    18 days later
    • [deleted]

    Tom, I just popped into here to tell you your last post was bang on its good to see someone with some fundamental understanding of key philosophical ideals such as infinity, zero, one , momnotheism/monism etc. that are paradoxial and relative and problematic. I am Agnostic and so Im a Deist and any discussion involving the combination of math or philosophy topics will always fascinate me.

    Bubba Gump, also your post was excellent and I agree that , in the end, religion and math and philosophy and science and numerology and chemistry and astrology and biology and even history may well all inevitably be the same thing hhehehehe. It truly is esoteric and too deep to grasp when looking for absolute answers.

    Seriously though, I am soon to write a 4th installment to my paper about paradox and what can and cannot be officially stated when it comes to proving which branch of math is more accurate than the other. It's most likely a tie and relative like Celsus vs Fahrenheit or geocentricism vs heliocentricism.

    Excellent posts

    • [deleted]

    I really should spellcheck first hehe. I meant to type monotheism...

    anyways, astrology, for all intents and purposes, may aswell be astronomy or math or science or biology or chemistry etc . etc .etc Seems we are obbsessed with patterns and we seem to think we can completely know these patterns and thus become like gods with predictions and knowledge ... but we can't ... we are only human.

    • [deleted]

    Some of you may think I went overboard and too deep with my relative thoughts about how everything is all part of a universal and fundamental pattern ... well this may help you see the universal patterns and connections that all these have with eachother;

    http://www.hiddenmeanings.com/body.html

    Too many questions, too many topics ... but still all fun to discuss

    3 months later

    The human term 'number' and the concepts of a counting system are descriptions of difference between topologically whole areas. 'Two fish' decribes two discreet entities within a set 'fish'. What we call number theory is the detailed analysis of how areas of difference within topologically whole entities organise efficiently within that entity.

    The differences described however are not the result of human numbering, human numbering is a classification of already existing areas of difference within a given set. A number of fish existed, in an awful lot of discreetly different ways, before the human number system. If we insist that the different areas only existed as areas of discreet difference after they were perceived to, we are what is commonly termed 'creationist'.

    It is accepted that the universe (by definition) is a topologically whole entity. Physics is the analysis of the areas of disceet differences, and how they interact, combine and divide within the topologically whole universe. In physics these areas of difference, and the way they 'organise' are treated as the results of naturally-occurring phenomena. Physics has always used mathematical tools to analyse these 'physical' areas of difference, and many words have been written about the miraculous coincidence that the language of mathematics is so well suited to do such analyses.

    but instead of numbers being miraculously suited to describing the universe; what WE call number is how the universe 'describes' its differences.

    the relationship between the 'naturally-occurring areas of discreet difference in the topologically whole universe, and their behaviours' and 'human numbering system, number theory and mathematics' is the equivalent of the relationship between 'the naturally-occurring force between masses' and what we call 'the theory of gravity'.

    relationship N->n

    equivalent to

    relationship G->g

    where the capital letter represents a natural phenomenon and the lower-case represents the human analysis of the natural phenomenon.

    The implications are that the naturally-occurring processes that we call 'number theory' will result in the naturally-occurring processes that we call 'quantum mechanics' and further to all other naturally occurring processes that we eventually call 'physics'.

    If the universe IS a topologically whole entity, and everything within that universe is composed of various fractions of the whole: then inflation is in fact division and subdivision. The expansion is in the 'numbers' ie the discreetly different areas within the whole.

    it is not a set of sets, which is then a set of set of sets... the set of sets is absolute by definition and any introduction of further sets merely shows subdivision of the original.

    [inserted note for Prof Schiller, with added lolz --> the term 'discreet difference' is used to indicate that although there may well be a continuum of difference it's only when such differences are discreet that they interact as differences. i love my analogies, so think of a magnet. there is a continuum between N and S (the physical object is a whole unit), and the differences in polarity gradually converge to the grey areas where we can't tell if it's more N than S or more S than N... but when the interactions of each pole are examined, we see they act in discreetly different directions. The continuum isn't discreetly different, so it isn't analysable through number. As soon as we're analysing using number we're separating it into discreetly different interactions. A curve on a graph is a continuum, but as soon as you wish to examine the value of a point on that line, you are separating it discreetly from the continuum of line before and after./note for prof schiller]

    It is eminently testable as it predicts that 'number theory' and 'quantum mechanics' will become increasingly converged (ok, all areas of physics... but I say quantum mechanics because it's at the narrow end of the decreasing complexity).

    the prediction is: more and more 'coincidences' such as the riemann-zeta function will be 'discovered' at the LHC and other high-energy early-universe particle experiments. (In fact anywhere all naturally-occurring topological wholes being subdivided over time, when analysed mathematically should show evidence's of 'strange' similarities between each other, whether it's in physics, biology or any other field).

    still with me?

    :P

    [oh... and if space, energy and matter really are just expressions of naturally-occurring mathematical functions governing the discreet fractions of a single existence... then shouldn't there be a new unit of existence? how about: Subatomic-To-Universal-Functions ... :D ]

    The human term 'number' and the concepts of a counting system are descriptions of difference between topologically whole areas. 'Two fish' decribes two discreet entities within a set 'fish'. What we call number theory is the detailed analysis of how areas of difference within topologically whole entities organise efficiently within that entity.

    The differences described however are not the result of human numbering, human numbering is a classification of already existing areas of difference within a given set. A number of fish existed, in an awful lot of discreetly different ways, before the human number system. If we insist that the different areas only existed as areas of discreet difference after they were perceived to, we are what is commonly termed 'creationist'.

    It is accepted that the universe (by definition) is a topologically whole entity. Physics is the analysis of the areas of disceet differences, and how they interact, combine and divide within the topologically whole universe. In physics these areas of difference, and the way they 'organise' are treated as the results of naturally-occurring phenomena. Physics has always used mathematical tools to analyse these 'physical' areas of difference, and many words have been written about the miraculous coincidence that the language of mathematics is so well suited to do such analyses.

    but instead of numbers being miraculously suited to describing the universe; what WE call number is how the universe 'describes' its differences.

    the relationship between the 'naturally-occurring areas of discreet difference in the topologically whole universe, and their behaviours' and 'human numbering system, number theory and mathematics' is the equivalent of the relationship between 'the naturally-occurring force between masses' and what we call 'the theory of gravity'.

    relationship N->n

    equivalent to

    relationship G->g

    where the capital letter represents a natural phenomenon and the lower-case represents the human analysis of the natural phenomenon.

    The implications are that the naturally-occurring processes that we call 'number theory' will result in the naturally-occurring processes that we call 'quantum mechanics' and further to all other naturally occurring processes that we eventually call 'physics'.

    If the universe IS a topologically whole entity, and everything within that universe is composed of various fractions of the whole: then inflation is in fact division and subdivision. The expansion is in the 'numbers' ie the discreetly different areas within the whole.

    it is not a set of sets, which is then a set of set of sets... the set of sets is absolute by definition and any introduction of further sets merely shows subdivision of the original.

    [inserted note for Prof Schiller, with added lolz --> the term 'discreet difference' is used to indicate that although there may well be a continuum of difference it's only when such differences are discreet that they interact as differences. i love my analogies, so think of a magnet. there is a continuum between N and S (the physical object is a whole unit), and the differences in polarity gradually converge to the grey areas where we can't tell if it's more N than S or more S than N... but when the interactions of each pole are examined, we see they act in discreetly different directions. The continuum isn't discreetly different, so it isn't analysable through number. As soon as we're analysing using number we're separating it into discreetly different interactions. A curve on a graph is a continuum, but as soon as you wish to examine the value of a point on that line, you are separating it discreetly from the continuum of line before and after./note for prof schiller]

    It is eminently testable as it predicts that 'number theory' and 'quantum mechanics' will become increasingly converged (ok, all areas of physics... but I say quantum mechanics because it's at the narrow end of the decreasing complexity).

    the prediction is: more and more 'coincidences' such as the riemann-zeta function will be 'discovered' at the LHC and other high-energy early-universe particle experiments. (In fact anywhere all naturally-occurring topological wholes being subdivided over time, when analysed mathematically should show evidence's of 'strange' similarities between each other, whether it's in physics, biology or any other field).

    still with me?

    :P

    [oh... and if space, energy and matter really are just expressions of naturally-occurring mathematical functions governing the discreet fractions of a single existence... then shouldn't there be a new unit of existence? how about: Subatomic-To-Universal-Functions ... :D ]

    22 days later
    • [deleted]

    There is something that smells in Denmark when it comes to adding QM and GR 2+2=4 In tandem.

    Just look at what happens when we add another drug to paracetomal to enhance it's effects.

    It results in halucinations.................

    Maybe the same thing results when we are doublemnined about physics insisting that both QM and Einstein are right and they can be combined...........

    Would appreciate your feedback guys you seem much more rational than the guys on the Hawking forum

    • [deleted]

    I have a provisional patent on this vrtual time TM clock and Casio R&RD UK are looking at basing a new product on it.Attachment #1: clock2.zip.zip

    2 years later
    • [deleted]

    Since the beginning of human language and civilisation we have gradually developed a formal language called mathematics. All human natural languages are intimatly coupled with our sophisticated imagination. They only have a limited self-contained logics. Mathematics has gradually been simplified and formalized. It is a language that has been optimized to expressed relations in the simplest possible ways. A driving force in the evolution of this language has been the simplification in the expression of the relations existing mathematical structures.

    There are many physical realities which have been discovered based exclusively on the assumption on equivalence a certain physical structure to certain mathematical structures. But the success of mathematics is mostly concentrated in the large scales and the small scales of the universes. At the biological scale and at the planetary scales the entities of interests are not equivalent to mathematical structures.

    For the ancient and modern platonists, the intimate connections between some aspects of the world and the mathematical world is taken as a proof of the objective existence of the mathematical world. The inventions of mathematicians are discoveries of eternal realities. Modern cosmologists by positing the a priori existence of mathematics and of quantum theory and general relativity as initially existing prior to the big bang are all platonist dualists.

    It is possible to imagine a creation scenario which would begin with the co-creation/ co-evolution of mathematics and the universe. The laws of nature would gradually evolves with the evolution and stabilization of the relations. Evolution would consist in the gradual explicit embodiement in physical structure of a subset of the regular implicit relations of the world. The platonic world being gradually constructed in the structures of the world.

    It is a gradual passage from culture/process (implicit relation) to nature/structures (explicit embodied relations).

    In this scenario all the layers of nature correspond to the appearance of a structure which accelerate the culture/nature transfer speed. The passage from biology to human culture correspond to the culmination of the evolution of mammalian dreaming into human consciousness which is a splitting of the awareness field between external and self-awareness (self-controled induce dream). The implicit embodied platonic wold which is the human bodied can then become explicit and able to be express in natural language and mathematics.

    19 days later
    • [deleted]

    In my opinion, it is a very hard choice to make between perceiving mathematics as evolving along with the universe and perceiving mathematics as what you called "supranatural" or beyond spacetime.

    To stand by the former would mean that we see mathematics as inherently part of the universe - that is, universe somehow has a mathematical and physical side at the same time. They are dual in the sense that as the physical universe evolves in certain ways, the mathematical universe follows suit such that there is always a correspondence between physical objects (matter, energy, etc), physical laws and mathematical objects (numbers, operators, spaces etc) along with their operations (addition, multiplication etc). This opinion itself is probably difficult to argue for or against, for lack of evidence to show that indeed the world is mathematical in nature, i.e. this matter is inherently a philosophical issue that most likely cannot be settled by means of experimentations or tests.

    To stand by the latter view would mean that mathematics is beyond our universe although from ontological perspective, it is existent. To explain the interaction or even how mathematics work in view of physical universe is harder by means of this stand, since we do not know how mathematics comes to be applicable to reality at all. However, it does make it easier to swallow the idea that mathematics does not need to produce physically feasible results/statements.

    The latter stance still mean we are facing the same problem as the former - the status of mathematics as being supranatural would be quite unprovable physically and hence in the face of science, it most likely cannot gain a place. Both interpretations require philosophical considerations.

    Pertaining the feasibility of applying simple mathematical rules to Planck scales, let us consider a Kantian perspective: that humans do have 'categories' imposed upon our experiences like rose-tinted spectacles. For example, Immanuel Kant postulated that space and time are both structures imposed by our minds to make sense of whatever sensory input we obtain - they rearrange and organize our perceptions. Let us suppose that mathematics is in this particular case adopting similar role. What can we gain from it?

    I believe what we gain is the dissolution of your question. If mathematics is a default structure imposed by our minds (this is different from claiming mathematics is an invention) - that addition and multiplication is a method to organize our perception on quantities, shapes, transformations and abstractions, then if we suppose there exists a human who can experience an instance of Planck scale observation or even thought experiment, he may in attempt to make sense of what it means to be at Planck scale try to come up with a formalism that addresses the problem. This is analogous to how physicists try to search for quantum theory of gravity: the physicists realize they cannot make sense of certain phenomena by brute-force combination of quantum physics and general relativity, and thus in that state they are unable to do anything concrete to the phenomena - just like how our hypothetical thinker faces the Planck scale dilemma. Upon meticulous thoughts over time, we hope physicists can resolve the problem, and thus we can hope the same for this hypothetical thinker.

    What is the point of the thinker above? It serves as an example that a third interpretation is possible: that mathematics is a mental structure of the mind that organizes perception. If this interpretation is chosen, the question "whether simple arithmetical operation is applicable" is a little bit weird in view of this interpretation: for unless there is something for the thinker to test that view, otherwise the hypothetical thinker cannot even decide the answer to the question.

    If mathematics truly evolve over time, I think we will be faced with big trouble. That means mathematical statements are all time-dependent, and regardless of how little the effects of time on those statements, now mathematicians have to find relationships between mathematics and time functions. This will, in turn, reduce mathematics to physical science - because mathematics cannot work by virtues of logic and a set of axioms anymore. Mathematics that depends physically on time will make it a physical science in our current definition of science as a field of knowledge, and this epistemological impact is, in my opinion, self-defeating because mathematics was meant to abstract things out of physical sciences and crystallize them to generalize many issues.

    Supranatural mathematics poses no such problems, but it does provide another set of epistemological issues: for example, how would we account for countless instances of highly close correspondence between mathematics and reality? For example, how can we deny certain physical statements because mathematics show them to be so by means of proofs by contradiction? All these issues are not any less thorny.

    I would prefer my third interpretation, though admittedly it has its own set of problems too. But I sure like the idea that mathematics may start to work only after it becomes computationally meaningful. Maybe someone would like to critique my ideas too?

    2 years later

    I think maths just describes the structure of transcendentally ideal space and time in which all phenomena appear.

    https://www.academia.edu/7347240/Our_Cognitive_Framework_as_Quantum_Computer_Leibnizs_Theory_of_Monads_under_Kants_Epistemology_and_Hegelian_Dialectic

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