Has anyone pointed out that nature itself is an intellectual construct?  Not all cultures have had the ancient Greek concept of phýsis, Latin natura, English nature. Therefore, if "nature" is one intellectual manipulation of reality, "mathematics" is yet another.  The relationship between the two should hardly be surprising, since both are mental "inventions." Manipulate reality a bit, and you get nature. Manipulate it a little bit more, and you get a mathematical nature.  

My personal opinion in all these question depends on how we can define the words "Nature, prove, mathematics" so on. I know I am repeating Wittgentstein. But I think that there are too many vague questions these days. Too many vague notions are drawn for funding easiness. Also, I feel that we the scholars are loosing integrity for the sake of research funds, recognizability, and so on. 

Thank you for your contributions.  Do you not think the mathematics of apples is distinct from the mathematics of space or water, and these different entities possess different mathematical behaviours and so mathematics is a behaviour not language?

The problem is complex and addresses to all applicability, explanatory power and creation of mathematics. Regarding creation, some of the mathematical concepts are built by basing on a primary reference to nature, in a naturalistic sense (as example, the notions of Euclidian geometry); so, although abstract, these concepts contain in some sense of extension natural physical things; the problem is whether more complex objects and structures have such correspondence in nature; mathematics is autonomous, self-applicable and self-generating and this could be a problem in proving (if possible) that all mathematical entities have correspondents in nature. Epistemologically, mathematics can be created ex nihilo (the simple postulation of a set of axioms creates a mathematical subject). Perhaps you will find useful the works of van Fraasen, Teissier or Ye regarding perceptual mathematics (for instance, Teissier has a thesis on how humans create the concept of straight line through a vestibular perception and a visual perception). Regarding applicability, things are more complicated, but still related to creation. The major problem is the justification of the huge rate of success of the application of mathematics in science and physical reality, what physicist Eugene Wigner called in 1960, "the unreasonable effectiveness of mathematics". The central problem is to establish the right kind of relation between the abstract domain of mathematics (subject to traditional ontological debates) and the physical domain of the modeled problem, through which to infer such justification. The first solution was given by Frege, through his semantic applicability of mathematics (mathematical objects are applicable to concepts and concepts can be instantiated in physical objects, under the second order logic). More recently, structural theories has been given basing on the principle of an isomorphic/homomorphic relation between structures of the mathematical theories and structures embedded in the physical systems, in a set-theoretic framework (basing on the principle that sets as mathematical objects can contain physical objects). Such external-relation theories (Pincock, Bueno & Collyvan) are challenged today to answer objections like circularities (see https://www.researchgate.net/publication/261855773_CIRCULARITIES_IN_THE_CONTEMPORARY_PHILOSOPHICAL_ACCOUNTS_OF_THE_APPLICABILITY_OF_MATHEMATICS_IN_THE_PHYSICAL_UNIVERSE  ) and the transfer of the truth conditions from the abstract to the physical domain (or the alteration of the mathematical necessary truth in the mixed statements with both math and physical terms in internal-relation theories). Mathematical explanation is part of the applicability problem, as explanation is one of the functions of a model (see classical examples such as Baker's cicada insects having primes as life cycles in years or bees’ honeycomb geometry for debates on what would make an explanation genuinely mathematical). The problem of the justification of the applicability of mathematics is open and we have to also consider degenerate theories or even semantically trivial reformulations of the problem: Is mathematics - as science, method or language - indeed applicable to a physical domain? Is it applicable anywhere or it is part of the investigated object or of the cognitive language by which we investigate it?

Article CIRCULARITIES IN THE CONTEMPORARY PHILOSOPHICAL ACCOUNTS OF ...

Thank you sincerely for you resonse Catalin.  For me, as you say, structure and isomorphism is evidence of mathematics in Nature. This is the point I try to make above when i mention apples and water, and say their mathematical behaviour is distict. Also I like your reference to Euclidean geometry.  I would say Pythagoras is the behaviour of space (locally).  And that human modelling of Nature using mathematics, gives the impression that mathematics is an invention; and the reason for that is that in instances of modelling, mathematics invented for the situation.  

Let's try a sort of philosophical exercise from what we already have in this interesting exchange.

On the one side we see Stefan thesis, according to which mathematics is a kind of language, a separate domain from physics "completely different ontological categories" with his own words. A very respectful and classical position.

On the other side there is Catalin's thesis, which, through the Euclideian example, shows us that "concepts contain in some sense of extension natural physical things", and opens to the challenging question of  "the unreasonable effectiveness of mathematics".

I would say that, even considering mathematics just an abstraction, this does not mean it can be really created ex nihilo; to abstract something is always to pull something out of something else (in latin ab-trahere means that). So, if we agree mathematics is an abstraction we need to ask ourselves where have we abstracted it from, we can therefore follow two different paths: we can say it has been abstracted from the physical world, so mathematics is a real feature of physics, since it directly derives from that by means of our abstraction; or we can say it has been abstracted from our mental world (a sort of hypersubjectivistic perspective), and in this case mathematics can be seen as a description of our mental architecture. In both these cases mathematics is not a conventional creation, or at least not just that, but it says something about something. The way it says could be ours, but what it talks about (the referent) is not.

Back to the topic, I think that, if we want to show the mathematical structure of the world to someone, concepts like isomorphism or even more analogy could be useful, but not conclusively, because your speaker should be previously persuaded that recurrence of mathematical pattern is a prove of mathematical nature.

I like to finish this exercise with this consideration, the mathematics of nature looks like the nature of mathematics.

Ciao

There is a more fundamental point: nature is inherently lawful because everything that exists has a specific nature and what a thing is determines what it can or will do--the law of identity--if all does reduce to math that is why-

Thank you Giampaolo for your very interesting little essay.  And thank you Edwin.  Giampaolo, I especially like:  "recurrence of mathematical pattern is a proof of mathematical nature." 

My research looks at arithmetical information in quantum theory.  I have spent more than 8 years on this now.  The central question of my research has been about arithmetic and the logic of quantum experiments. But as well as answers to that question, I have found that formulae (propositions) in arithmetic that are theorem of arithmetic's axioms have something to say about what we witness in Nature. As time has gone by I have become more and more convinced that Nature is highly arithmetical -- to the extent that I take seriously the possibility that physics stems from arithmetic.  

My interest in asking this question is not philosophical.  I am interested in making progress with physics.  And so, in my original question, in asking for evidence that might prove, I am interested to know what physicists might accept as proof.  What kind of evidence would convince physicists? 

Science is never free of philosophy even if they ignore it--metaphysics and epistemology underlie all science--is there a reality out there? is it knowable? by what means? Kant, by denying that reality is knowable, did more damage to science than any scientist ever did-

I don't know of any philosophy who would deny that Kant's central thesis was that reality, the noumenal world, was unknowable and that we can only know the phenomenal world based on inborn categories which make "reality" conform to our minds--so you see everywhere today that science is only a science of appearances--now you are right that there is another influence at work: Aristotle who believed that reality is knowable by the senses and reason--good science has always been due to his influence (not to mention the Enlightenment)--you can say that this is a battle to the death but generally Kant has been more influential in the last 100 years--but we owe our lives to Aristotle--

Yes but science for him has to be  a science of appearances not of reality--experiences are not of reality--they are determined by the built in categories of the mind--BTW: Aristotle was probably the first scientist--he founded the field of biology by simply examining (and dissecting) organisms--the experimental method was not known then, however--that may have come from Bacon or Mill-newton used that method to understand the nature of white  light-

Here is a Kant article FYI: Postmodernism's Cognitive Roots by Onkar Ghate in my edited book: Postmodernism and Management: Pros, Cons and the Alternative, JAI, 2003. It answer your question and shows why Kant's demand, cognition without a means, is irrational. As to Plato he critical error was concepts without percepts, making concepts totally divorced from reality--we call them floating abstractions--and  thus subjective. The ch. BTW. has many quotes from Kant. I may have a copy of this--see attached-

The Kant attachment is only a summery statement not the chapter- I do not have the full chapter on my computer but it is in the book-

The book chapter makes the issue of what Kant stood for totally clear- and ghate has read Kant and quotes from him- so I will not discuss this any more until you have read it- sorry I do not have it on my computer -Kant did not just destroy reason- he also destroyed values in a very vicious way-this you can find in L. Peikoff The Ominous Parallels- you can find more in his book The Dim Hypothesis in which he shows how Kant was the great destroyer in the history of philosophy

Edwin,

Kant was a total supporter of the physics of Newton.  He did not beleive that it describes reality as it is but he beleives that the equations were right.  And we now know that the equations of the mechanics of Newton have limited application.  We know that we will never reach a physics that will perfectly describe reality as it is and reach the end of science.  We now consider scientific theories as limited truth, not as absolute truth.  And quantum physics shows that the state of a system cannot be entirely known in principle.  Here again an intrinsic limit to the penetration of knowledge into reality.  To accept that our knowledge will always be partial, limited is not denying science nor even limiting science. 

I. Kant, K.R.V, der transzendentale Idealism als ...:

"Wir haben in der transzendentalen Ästhetik hinreichend bewiesen: daß alles, was im Raume oder der Zeit angeschauet wird, mithin alle Gegenstände einer uns möglichen Erfahrung, nichts als Erscheinungen, d.i. bloße Vorstellungen sind, die, so wie sie vorgestellt werden, als ausgedehnte Wesen, oder Reihen von Veränderungen, außer unseren Gedanken keine an sich gegründete Existenz haben".

I've always been struck by this Kant's statemen, where he says that we can just know phenomena and these phenomena are nothing but our representations.

Kant has for sure a great place in the history of philosophy, and therefore of science, but he is also responsible of the idealistic turn which affected modern thought since Descartes' time. He's no different from Descartes in that respect and he could not build the bridge between thought and reality, since he was the one who divided the those two realms since the beginning.

To put it in contemporary words: kantian philosophy lacks of reference, as Frege will tell us, you can find objects, meanings maybe, but not reference in it.

Back to the main topic, I think Kant is of little use in a discussion which aims to persuade someone of the intrinsic mathematical nature of reality, since Kant himself would be uncomfortable in a topic like that. More precisely, Kant probably would label as noumenical any kind of argument about "Nature", and therefore unknowable.

Besides greek philosphers like Pythagoras and Plato of course, another crucial western thinker who made a statement like that was Galileo, who wrote in his master piece, il Saggiatore:

"La filosofia naturale è scritta in questo grandissimo libro che continuamente ci sta aperto innanzi agli occhi, io dico l'universo, ma non si può intendere se prima non s'impara a intender la lingua e conoscer i caratteri nei quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi ed altre figure geometriche, senza i quali mezzi è impossibile a intenderne umanamente parola; senza questi è un aggirarsi vanamente per un oscuro labirinto.”

In English it could sound like this:

Natural philosophy has been written in this great book which stands wide open in front of us, I mean universe, but it cannot be understood if we don't learn the language and the characters it has been made of. It has been written in mathematical terms, and characters are triangles, circles and other geometrical figures, without which it is impossible to make any sense out of it; without these it is an empty route through a dark maze" (I apologize for my poor translation, consider it has been written in an ancient and a little baroque italian).

Before closing this already too long reply I'd like to drive your attention on the fact, which I find curious at least, that even Galileo, after such a statement, declared his distrust on the possibility to reach the essence of reality, focusing therefore on the secondary qualities (the phenomenical world). I find this particularly strange, since his faith in mathematics as language of nature is not easily compatible with this final skepticism: what is mathematics in his words, if not an essence?

Ciao

Giampaolo (and othre Kanitians)

I presume you mean: Kant’s « Critik…” (1781/1787) A491/B519?

Stefan: the same, A51; B75

(… everywhere on RG, the same, just the same…)

Dear friends,

I just discovered this thread. Therefore, as Stefan asked, let us try to have right citations, for the start. Then we can develop further. I have to read all the posts to understand where the problem is. As so often, it may be essential misunderstanding of some Kant's concepts, in particular of transcendental and, a priori, and in particular noumenon, i.e. Ding in sich.

The difference between aritificial mathematical system and mathematical nature is only one that there is a Turing transmit function delta as instructions to run a artificial system, but no such instructions expressed any where in cosmos! Do you once read a natural book of instructions wrote by natrue ? You read Newton theorems expressed by human rather than nature itself.

So "natrue is mathematical" is a psuedo proposition!

To all

The point of my yesterday's comments was that we should here on RG try to use less fast Internet knowledge to learn about the problems and pretend to know and win the discussions. Instead we should slowly start to use more basic original works. The problems will thereby not be “solved” or disappear; they will become even greater. Yet they will be much deeper.

Thank you Dragan.

I am fairly new to this site.  I am very impressed with the level of discussion in comparison with elsewhere, but there is slight tendency for some people to try to win discussions.   This idea of winning blocks the free flow of ideas.  

Well, KRV stands for Kritik der reinen Vernunft, I quoted even the chapter, it's not an internet cut and past, even if you can easily find it on every common research engine, the same goes for Galileo's quotation. Both of those passages have been taken from primary bibliography, I don't really understand the objection about "internet knowledge".

More broadly, as much as I love to discuss issues related to the history  of philosophy, this topic was not born to ponder that kind of matter, but to consider a theoretical problem, a very important one. Kant or whoever else could be useful interlocutors, but they are means, not the ends of this question, that's why I would invite whoever wished to contribute to the topic to stick on it, not indulging in interesting but collateral themes.

Dear Giampaolo,

I am sorry. I only insist on the points that may improve on understanding and on locating the citations. Your citations, I think, were very well chosen and most of the comments were standard interpretation of Kant’s ideas. Yet you put something like this:

“I.Kant, K.R.V, der transzendentale Idealism als ...:” instead of I. Kant, KrV, B519 / A491 or even you could be more specific with adding 26-33. The system A and B for the first and second edition has been used since the edition of the “Philosophische Bibliothek” (1868, i.e. 1926 etc., now available in the version from 1956 and later by W. Weischedel). The best edition is for me this of Meiner, by Jens Timmermann The chapter that you gave is not o.k.: you gave an incomplete and false title in fact. For the majority it would be hard to find.

Please go to your book and see the title of the chapter again and you will see. You need to say that it is part 1, chapter 2, book 2, division 2, section 6 ... (quite absurd of course) and not forget the coma… in the subtitle. On the internet we have unfortunately a lot that is false. BUT: I insist that people do not try to be smart and do fast Internet search and provide answers ON ANY POSSIBLE question that could be imagined, even which is absolutely out of their expertise. This introduces tremendous confusion and permanent ambiguity on RG and makes particularly philosophical discussion IMPOSSIBLE.

You say “I've always been struck by this Kant's statemen, where he says that we can just know phenomena and these phenomena are nothing but our representations.” Implying that Kant thought that world was just our imagination! This made me respond to the trivial mistakes in your reference for your citation.

To go back to Steve's question. If we continue in Kantian direction, then discovering “noumena” and comparing them to mathematical descriptions that we already have about them, would be enough to confirm or reject that mathematics is already there, inherent to nature. Kant implies that this could not be done. I think that  the phenomena without their properties would become "invisible". Kant thinks that we would need some "other" method to see them...

But in fact, I did not want to disturbe you so much..

Dear Dragan,

I am sorry too, I confess I still don’t see this big deal providing a quotation which everyone could have easily checked out, without giving the proper reference to the critical edition. I found the passage I needed on my edition of Kant’s work (the one edited by Wilhelm Weischedel, Werke in sechs Bänden, Insel Verlag, Wiesbaden 1956), and then I’ve looked for an on line source to make it quickly available to those who could have been possibly interested. In fact, if you google “der transzendentale Idealism als” you find the whole quotation and its context. The title I gave was enough to reach the source, this is not a paper to be published in Kant-Studien Journal, it’s a forum. (That title might have been incomplete, in a philological sense maybe, but why false?).

Please, be patient, if you imply from what I’ve written that Kant thought the world is just our imagination, this is your guess, not mine. I imply what I’ve written, quoting Kant, nothing more nothing less. The problem is the knowledge of the world, not its existence. It’s a total different problem, which is unsolved in Kantian terms in my opinion, together with many others’. We can disagree on Kant’s interpretation, but we should understand what we disagree on, before starting to blame our interlocutor.

Dear Stefan,

I am glad you appreciate GG thought. Your grammatical analysis is interesting. Allow me to keep on going that way. The passive form was undoubtedly a smart move, still I wouldn’t go that far deriving the hypothesis you drove, GG just left unanswered the question about who wrote that book, but he made clear the language it was written by, which is the point we are discussing. We can skip Plato’s or Kant’s approaches (just for the sake of pedantry I’d add that in many philosophical historiographers Kant stands on the line started by Plato), God, or we, to ponder the medium: the language. Of course this is a platonic idea again expressed in Timaeus (maybe Schleiermacher was right when pointed out that all the western philosophy is a Plato’s work footnote), but for our topic it’s worth to note that the language in this conception is not divided from the realm of being, it is a form of being itself. My suggestion would be to closer consider the notion of number, not the abstract cardinal one, like 1, 2, 3 ... but the ordinal one, especially as Greeks thought of them, arithmoi-logoi like double, half, triple ... these relationships are inherent to the realm of being, of course we need an intellect to spot them, but to spot does not mean to create ex nihilo.

We may have invented the natural numbers to count objects, but then prime numbers were discovered, not intentionally created. In other words, some math might have been forged by human mind, yet we humans have found in it way more than we had initially put. And it is curious how prime numbers, apparently of little use, are at the bases of the solutions of concrete and current problems like those of cryptography and digital security.

Ciao

Giampaolo

I did not want to discuss more profoundly the Steve’s question but just to make formal objection to the way how we discuss on RG, and not only on this thread. There is another thread on the same question where I already wrote more than 300 comments so I have little to add now.

https://www.researchgate.net/post/Is_mathematics_a_human_contrivance_or_is_it_innate_to_nature

After discovering your profile and that the subject discussed here is a part of your broad professional interests, I am even more disappointed by the way how you gave the reference to Kant’s work. In addition, sorry, just another objection: when you give an introductory paragraph from Kant’s text but do not give more text from the same page (!), where it is shown what he really thinks, and do not explain what exactly you wanted to illustrate, you may mislead a reader. The other points that you mention, are of course fine with me and a valid arguments of discussion.

All the best.

My reason for asking this question was a practical one.  My work looks at physics from the viewpoint of ordinary arithmetic, as a formal axiomatised system.  I show there is (well-known) logical independence, inherent in arithmetic, that conveys into quantum wave packets, prior to measurement. And that that logical independence is located where quantum experiments are indeterminate.  This would indicate to me that arithmetical processes are going on in Nature.

And so I am interested in practical ideas that might actually confirm the mathematical nature of Nature. 

Steve

Not to misuse again what Friedrich Wilhelm Joseph Schelling wrote at the end of his Foreword to “The System of Transcendental Idealism”, let me just admit: There are many threads on RG that I cannot follow, most often because of my lack of knowledge; here, you convinced me that this thread also, is not for me.

@Dragan

What a pity.  But thank you for looking.  Actually, the ideas are simple; t's the jargon that is the barrier. 

Dear Steve,

I downloaded all your articles. if I'll get some leisure time, I will try to see about the “jargon”.

@Dagan

You might prefer to wait for the rather more definitive rewrite I'm working on now. I'm expecting it to be finished in 2 or 3 weeks.  I'd say it is less confusing.  Pleased you're interested.  Thanks.

Let me try to articulate a rather simple answer. Human is a product of billion of years of evolutive optimisation such that humans understand in a hight fidelity, nature. In this way nature is coherently reflected in human (in brain, sense organs, and body). On the other hand humans construct "vision space", "aptic space", etc. Finally humans construct mathematics which is based on this reflected real world. Thus "nature" as we perceive it, actually is a mathematical one. There are of course other expressions of nature: poetic, narrative, artistic, etc. But I think that "structure" is underline all these. And who knows, at some point in the future, science and art would converge to the same point.   

I am certain that the cave men believed that their bones throwing method and the positions that the bones displayed on the ground, gave faithful reflection of the real World.

Dear Drangan,

No offence being a Platonist and no offence either being an Aristotelian! 

Thank you Filippo.

I think you are saying that objects that seem to be discrete are actually not discrete (or may not be).  Would you not agree then, that you are saying that Nature is arithmetically like space or a fluid rather than like apples?  And would you not deduce, therefore, that you are making a claim about the arithmetical behaviour of Nature?  And that your claim, therefore, assumes Nature does possess arithmetical behaviour? 

May be, Steve, But, it appears that Flippo referes to the "things-in-themselvs". Let us see what he is going to answer.

Costas

I wrote it many times on the other thread. Why is this so difficult to grasp? How simple I should put it to be understood? let me try again.

Our theoretical models of reality depend almost entirely on the methods that we have to gather knowledge about nature. In antiquity, the four causes and the four qualities of Aristotle determined almost completely how he was able to see the reality. As we in 19th and 20th century developed not only many new methods but the measurement become quite precise, our theoretical models flourished. How will it be in 2.000 years? Well, we will be able to measure even more then we can now, will introduce some different mechanisms, causal mechanisms or even some other mechanisms of the interactions on distance, and… for us, world will have some new qualities. The “old” assumptions will not be completely wrong, as Aristotle’s heat and cold were also not some qualities from antiquity that we hold wrong today. Still we will have, we will construct!  different picture of the reality and will propose different theories – and, as Costas (ALMOST) wrote: “Finally humans will construct a mathematics which will be based on this (future), reflected real world”. Would the “real world” in the year 5014 be some different world? No. Our knowledge will change and we will construct a new, different world – as we CAN SEE IT.

The new mathematics? Yes, the old, our, mathematics, will not be wrong – it will be obsolete. But the new will still be - not as the real world IS; it will be such as OUR BRAIN can construct it.

@Stefan

I would say, we don't need to recognise mathematics in Nature for it to be there.

The ancient astronomers had conceived a mathematical model describing/predicting how the stars and the planets move relative to the center of our planet.  This model describes the phenomena it is supporsed to describe and allow prediction.  We now have another more general model.  The old model can still be used for predicting planet positions but it cannot be used as a model of the cosmos.  But all scientific theories are like this old Ptolemy model, they are limited partial description of reality from a very specific viewpoint.  We do not know the limit of the new models , we will know that when they are subsumed under new theories.  How intrinsic to reality was the old Ptolemy model? 

nature = fysical laws and laws = mathematics, so nature = mathematics.

The same answers again.

Would you please advance a different position AFTER you managed to reject, for example, one that I offered couple of hours ago.

I suggest that we avoid to make political declarations but advance the arguments that demonstrate that what I for example say should be wrong and what you want to say is more probable. Even if this would be something new for you, please try to use arguments, it is more fun to respect scientific method.

I am sorry, Steve, it appears that I am just disturbing friendly, relaxed atmosphere of the people interested in science. Please, you people, you do not have to follow what I suggested.

Thank you all.   I have come to the conclusion I need to post this question again, worded differently.  My premise had been that Nature is, indeed, intrinsically mathematical.  But I did anticipate that I would provoke a debate.  Of course, obviously, I should have.  My purpose is to gather evidence that can be used formally as a research base.

Steve,

The only evidences we have is that nature is extrinsically mathematic in the sense that our most precise extrinsic descriptions/theories are mathemacally expressed and are always expressed in terms of extrinsic measurement processes.  It is very usefull and even essential for us to do that but I am not convinced that nature is intrinsically mathematical.  What is mathematical is what is fixed or stabilized enough to be described approximatly by fixed mathematical model.  My intuition tells me that nature is at the core a flux (the opposite of stability) and that from it aspects of realities stabilized in almost fixed structures describable my mathematical models.

Has anyone pointed out that nature itself is an intellectual construct?  Not all cultures have had the ancient Greek concept of phýsis, Latin natura, English nature. Therefore, if "nature" is one intellectual manipulation of reality, "mathematics" is yet another.  The relationship between the two should hardly be surprising, since both are mental "inventions." Manipulate reality a bit, and you get nature. Manipulate it a little bit more, and you get a mathematical nature.  

Gruner,

Science is a kind of interactional interface with some aspects of Nature expressed into a scientific language (mathematics being the formal core of it).  All animals have an interactional interface with the part of nature they interact: their umwelt.  Their umwelt is defined by their body and more specifically their sensory-motor system.  A animal body is an instrument: an interacting interface with an umwelt.  Human are unique in having a sensory-motor system that can allow the extension of its body by adding to it an external interface: a stick, a car, a window screen.  While other animals have a single instrument , we are shape-shifter or multi-instrument animal, or animal with different interacting interface.  I will leave aside how our nervous system is able to extend our body in varying instruments and umwelts and explore the consequences.  The consequence is that we can build conceptual interface in include them into our body and interact with the world with them and these interfaces are our culture, science , mathematics. But it is simpler to see our interfacing constructions as reality in itself, the old tendency to deify our own creations instead of continuing creating better interfaces.

Dear Steve,

Your question makes me think about 2 things:

1, some people are really able to offer evidences needed to prove Nature is intrinsically mathematical, at the same time, many people are really able to offer evidences needed to prove that mathematics is from human mind: “discovered-or-invented?”

2，mathematicians are like painters, mathematicians’ works are similar to pictures by painters. Is a masterpiece by a painter “discovered-or-invented?”

Regards

Many people believe “Nature is innately mathematical”. But it is equally fair to say “Nature is innately physical” and “Nature is innately chemical”…. Mathematics is often overemphasized by its characteristics.

Is mathematics one of sciences? Sciences are built-in, so mathematics is built-in, too. I think this is relationships among Mathematics, Science, Philosophy and Nature.

Clearly Nature is not mathematics but without mathematics is impossible to have a science of Nature.

1. Their laws have to predict new phenomena or situations and this implies to write formulae able to do it.

2. Their laws must be reproducible, i.e. using certain conditions we can find certain results in every part or time.

3. But what can give a answer to your question is that once you have a system of equations (in principle only following quite abstract rules of mathematics) describing a physical phenomenon you can see if these equations are compatible or not.

4. Perhaps the most paradigmatic example are Maxwell equations. He took all the equations of electromagnetism of 19th century and observed that they were compatible among them if he changed the Ampere's law, because suddenly the conservation of electric charge arisen. But what was more important, he obtained wave equations for the electric and magnetic fields whose velocity was exactly the one that the opticians were measuring for the light.

5. Playing only with mathematics he could discovered new electromagnetic spectra hidden to the eyes such as infrared or ultraviolet one. And many other things that today everybody use in communications or in electronic technology.

We human cognize nature from different angles and different ways; so, we have different branches of science.All branches of science are our cognized products with their own characteristics; we appreciate the different characteristic of different branch of science.

It is understandable that different scientist is in favor of the typical characteristic of the engaged field; so, may be every branch of science could be intrinsically if Nature is intrinsically mathematical. 

Are things in mathematics invented or discovered?

   All branches of every Science of Nature must follow the Physics laws, even the brain of a mathematician thinks with electric interactions among the dendrites of his/her neurons. Mathematics has born as a necessary tool and no as a game. On the other hand, it is a product of many cultures or people along the history of the mankind who have applied the principle of usefulness or adaptation like the species have made in the evolution of life.

  Let me to put one question which is not so different than the last one for mathematics (from my humble point of view), are the first law of Newton or the genetic model of evolution creations or discoveries?

See my paper: "Gödel On Truth and Proof: Epistemological Proof of Gödel's Conception of Realistic Nature of Mathematical Theories and that Their Incompleteness Cannot be Proved Formally."  (2011, delivered at University of Pittsburgh 7th International Fellows  Conference of the Center for Philosophy of science, 2012).    

In many places, we can see so many topics discussing “things in science invented or discovered”. Actually, it is a fundamental problem of “philosophy of science” which has been confusing people all the time-------how much can we human do in human science: invented or discovered?

Discussions with different formal languages go on and on swinging between “invented” and “discovered”.

Of cause this may be good to review some knowledge and sometimes to attain some inspirations.

   Dear Dan,

   Can you summarize the information of your paper in relation with this issue? Thank you.

One way to gauge to what extent nature is intrinsically mathematical is to imitate the Pythagoreans and listen to the music of the spheres (musica universalis).

Pythagoras conceived the universe to be an immense monochord, with its single string connected at its upper end to absolute spirit and at its lower end to absolute matter--in other words, a cord stretched between heaven and earth.

http://www.sacred-texts.com/eso/sta/sta19.htm

The Music of the Spheres incorporates the metaphysical principle that mathematical relationships express qualities or "tones" of energy which manifest in numbers, visual angles, shapes and sounds – all connected within a pattern of proportion.

http://en.wikipedia.org/wiki/Musica_universalis

   If Mathematics were only a set of abstract rules without any contact with reality, then we could have a game like chess: beautiful, interesting but without usefulness. On the other hand, if we assume that it summarize rules depending of Nature and we can recognize logic with our mind made within the "reality", therefore it is logic that it has applications and also the parts without it are going to be forgotten.

   For instance, tensors are mathematical objects important because the Physics use them and generalizations as functors are nothing because no application was found.

Dear Dan Nesher,

I asked you to give a summary of your mentioned paper on Gödel. Frankly speaking I doubt that it has anything to say on this question. He solved one of the questions put by Hilbert at the beginning of the XX century, very interesting about the completitud of a mathematical system. But I am afraid that it is quite similar to the turtle approach of Aquiles which was solved with the infinitesimal calculus and the concept of limit. Today some people argues that it is impossible to find a Theory of Everything in Physics (ToE) for this logic theorem, but I am absolutely sceptic.

Perhaps you have another point of view on the subject.

Dear Daniel，

Do you think Zeno’s Paradox of “Achilles-- turtle Race” has really been solved with the infinitesimal calculus and the concept of limit?

    Dear Geng,

    Thank you very much for writing properly this paradox. I all that I know it is a turtle that you always divide by two the distance that remains and this is an infinite process. Therefore the logic tell you that it never is going to reach the goal, but the infinitesimal calculus prove you that this process has a finite limit and obviously it can be reached the goal and even go through it. Is there more than I do not know on this "paradox" between logic and mathematics?

Dear Daniel，

As is known even in Zeno’s time many people knew “the infinitesimal calculus proved that this process has a finite limit and obviously it can be reached the goal and even go through it”, the most convincing and easiest way is having a real race between a real turtle and a man. Zeno created this paradox to disclose the fundamental defects in classical infinite theory system. These defects are still being there unsolved------we can see from many papers that many people are still trying hard but fruitlessly.

Dear Geng.

Infinitesimal calculus started with Leibniz and Newton, many centuries later that greek Zenon's time. 

Dear Daniel，

Sorry, what I mean “the infinitesimal calculus” is the operations with the limit idea, not Leibniz and Newton’s calculus.

Dear Geng.

Perhaps you think in the Indivisibles method of Archimedes for calculating areas or volumes. It is very far of the Infinitesimal Calculus that it was introduced in XVII century, and what is important for our discussion it had the idea of limit of succession or serie. The idea of mathematical continuous belongs to Cauchy.

Infinitesimal Calculus was one of the key points of Mathematics and it belongs no to "spirit world" but to the real world were it is absolutely necessary for defining properly motion or force on an object. Thus Mathematics is one discovery as could be Physics or Biology, from my humble point of view. 

Dear Daniel，

In fact, we human have “infinitesimal” along with “infinity” since antiquity and we had “infinitesimal relating calculations” in many areas since then.

I am sorry that I should say “infinitesimal relating calculations” to avoid ambiguity.

Quoted from Charles Francis

"There is no limit to a limit" as the tortoise quipped to Achilles after the race, "since for any delta we can always find an epsilon such that delta is not small enough".

"But then", said Achilles, "all I will have to do is choose a smaller delta, and the result will be closer than your new epsilon". "In that case, answered the tortoise I will give you another epsilon, such that it is not". And so it was that even though Achilles and the tortoise managed to complete the race, they never finished the argument.

   Dear Geng.

   One thing is to speak about infinitesimal or infinite, and another very different is to calculate a limit and to give a number.

Dear Daniel，

As we know, we now really have 4 confusing “infinitude” related things in our science:

(1) potential infinity as something unknown, as the pre-Aristotlean Greek believed,

(2) actual infinity as a property of sets, as Cantor thought on it,

(3) infinitesimal with the number form of X--->0,

(4) infinity as big number, as the pre-K12 child think on it.

We have a big trouble:

for the first 2, we have long-drawn-out and ceaseless “potential infinity--actual infinity” debates since antiquity; while for the last 2, people use them as “numbers relating to infinitude” in many kinds of practical calculations but they are “non-number number of variables（"One thing" theoretically they should not be numbers but "another very different" practically they should be numbers）”.

What can we do?

Dear Geng.

What is fundamental for a mathematician is to find a rule to assign a number or calculate an expression. The idea of a limit of function is very far of the greeks because they didn't have anything of this kind and this is exactly the beginning of the infinitesimal calculus that I have told you from the beginning of our discussion.

Much more complex is to try to work with infinities and to distinguish the cardinality of the Natural numbers (infinite ordered set) and Real Numbers (infinite continuous set of numbers). The cardinality assigned by Cantor to the Natural numbers was called aleph zero while 2 raised to aleph zero give the following cardinality of the Real numbers. This is what is known as transfinite numbers, a very abstract part of Mathematics quite far of the Infinitesimal Calculus that I was speaking on my previous posts. Obviously this is very interesting and perhaps even more interesting for people related logic-mathematical systems.

Dear Daniel，

Looking back into our infinitude related science history; we can see clearly that limit idea came along with infinitesimal and infinitude.

Of cause one of an important work and technology for a mathematician is to find a rule to assign a number or calculate an expression and so on. But another important work and technology for a mathematician is to ask “how” and “why” with our mathematical experiences and to build the related theory system proving that mathematics has closed relationship with nature.

Zeno's paradox ifsreally very silly because we know that people do in fact cross the room. So if this is a mystery to mathematicians so much the worse for them. It might help to recognize that infinity, though it is used in calculations, only refers to a potential (i.e., to add one more number). But in reality, everything is finite.

Dear Edwin and Geng,

Choose a number as high as you want. Do you think that it is the highest? No, you always can add it another one. This is infinite and therefore from a pure mathematical point of view, this concept is logic and real within the set of numbers, for example. Obviously this is not the case for other set of real objects as cars, birds, etc.

This is the kind of differences which could lead us to think that mathematics is no a natural science but, from my humble point of view, I do not think so. What is value of the electrostatic energy in the center of an electron? It is same if we do not use other techniques as the renormalization group which is out of classical electrodynamics and nobody would doubt of the reality of the electric charge. 

Daniel,

What is realist in this concept of infinity is not infinity itself but the concept of a non-stopping process which can be realized but never its fullfilled realization.  Mathematical logic can posit the concept of infinite set realized by non-ending process as a logical structure but there is no equivalent in nature of a realized result of an non-ending process.  Since mathematic by the axiomatic method disentangle itself from any contact with nature and so posit a priori what is true for itself, mathematical construction are necessarily true in this self-constructed world.  But any conclusions in that world cannot be assumed to be true of our world after setting itself free. Mathematic by the axiomatic method set itself free of realism.  Any claim by mathematician that some aspects of nature are essentially mathematical is a contradiction of the very method of mathematics.  We have to find the reasons of the effectiveness of the mathematical language for the modeling of the world elsewhere than in intrinsic nature of reality.  

I think there is confusion deriving from our understanding of what an infinite set is -- when at the same time, infinity is defined in arithmetic, to not exist.These two notions must be kept separate.  If you think of a number, yes I can always respond with one larger.  That might imply an infinite set, but it does not imply existence of infinity as a number.

Steve: I think we agree about infinity.

   Let me to give you two examples of infinite in Nature: time and radius of Universe.

The examples that Daniel gives, are related to either IST Internal set theory) or to AST (alternative set theory) where the "unlimited" (a non Cantorian version of "infinity") is essentially extremely large incomprehensible sets!

See also my previous posts.  

(1) Theoretically, in present traditional infinitude system, the “infinitude” concept can be divided into “potential infinitude” and “actual infinitude”. But no clear, self-justification definitions of “potential infinitude” and “actual infinitude” can be found in our philosophy and mathematics so far-------that is why we have long-drawn-out and ceaseless “potential infinity--actual infinity” debates since antiquity;

(2) Practically, when we meet infinite things, we have no ideas whether the infinite things we meet are “potential infinitude” or “actual infinitude”-------- that is why we have suspended “infinitude related paradoxes families”.

Daniel,

Maybe the actual size of the universe is infinite; it is a possibility that is compatible with the current Big Bang theory.  But it is not a possibility that is testable. The case of time is more dubious.  Our only way to go back in time is to infer it from structural changes. If we go back 13.7 billiion years at that point there is very little structure left in the universe and it is unlikely that we can infer an infinity of time based on this small amount of physical structures.  Most likely, the very concept of time cease to have any meaning around that point so making time finite.

Dear Louis,

You are speaking of space-time associated to our Universe and as we do not know if it is open or closed (Riemannian or Lobachevskian) or its acceleration, everybody write the future in infinite. But you are right, that the past is finite.

Daniel,

You are right, I had totally forgot the future!!!  Maybe because I think that it does not exist.  Yes it exists as a possibility but this is not existence.

    Dear Louis,

    The future is so real as the past (if we use the concept of space-time associated to certain quantity of energy-momentum) or if you force me to go to this choice, even more than past. The past is not possible to be measured while the future it is, over all the closest one.

Daniel,

Any measurement process happened in the past at the time of obtaining the result, so it is about the past.  We cannot change the past.  It is totally stabled while the future does not yet exist and can be change by what we do now.  But it is not really change since it does not yet exist but will depend on what we do, but it will never exist because we are always now.  While the past is with us now because the past is the stabilized aspect of the now while the future now depend on our past and the what is now that is now yet stabilized or taken form.  The past, or the structure of the now, allows us to make with a certain probability what will be the structure of the futur but this prediction is only as good as our prediction model is good, i.e. our knowledge of the past.

Dear Louis,

We are speaking about different things. The laws of Physics are invariant under translations in time ( conservation of energy), therefore we know that one experiment is independent of the time that is going to be made. Unfortunately we cannot come back in time to repeat it, although the physical laws are not against it. The problem is the row of time: increase of entropy for a closed system, only expansion electromagnetic waves and expansion of Universe.

Dear Daniel,

The conservation of energy and the time invariance in my opinion did not hold at the beginning.  I am consistent with the idea of the evolution of the laws of nature with nature itself.  Conservation of energy or the symmetry of the laws of nature with respect to time means that nothing in those laws change.

I do not understand your last sentence. 

Dear Louis and Daniel,

If you divide universal time and space: past, present and future, we human are at present (middle). We are not sure when we human will be gone. So, if future after middle is infinite, the past before middle is equally infinite.

I am talking about absolute (objective) “infinite”; but if no human, no human science and everything in human science is gone then of cause no “infinite” and “finite” concepts.