“A majority of contemporary mathematicians (a typical, though disputed, estimate is about two-thirds) believe in a kind of heaven – not a heaven of angels and saints, but one inhabited by the perfect and timeless objects they study: n-dimensional spheres, infinite numbers, the square root of -1, and the like. Moreover, they believe that they commune with this realm of timeless entities through a sort of extra-sensory perception.”There’s no reference for the mentioned estimate, but what’s worse is that referring to mathematical objects as “timeless” implies a preconceived notion of time already. It makes perfect sense to think of time as a mathematical object itself, and to construct other mathematical objects that depend on that time. Maybe one could say that the whole of mathematics does not evolve in this time, and we have no evidence of it evolving in any other time, but just claiming that mathematics studies “timeless objects” is sloppy and misleading. Holt goes on:
“Mathematicians who buy into this fantasy are called “Platonists”… Geometers, Plato observed, talk about circles that are perfectly round and infinite lines that are perfectly straight. Yet such perfect entities are nowhere to be found in the world we perceive with our sense… Plato concluded that the objects contemplated by mathematicians must exist in another world, one that is eternal and transcendent.”It is interesting that Holt in his book comes across as very open-minded to pretty much everything his interview partners confront him with, including parallel-worlds, retrocausation and panpsychism, but discards Platonism as a “phantasy.”
I’m not a Platonist myself, but it’s worth spending a paragraph on the misunderstanding that Holt has constructed because this isn’t the first time I’ve come across similar statements about circles and lines and so on. It is arguably true that you won’t find a perfect circle anywhere you look. Neither will you find perfectly straight lines. But the reason for this is simply that circles and perfectly straight lines are not objects that appear in the mathematical description of the world on scales that we see. Does it follow from that they don’t exist?
If you want to ask the question in a sensible way, you should ask instead about something that we presently believe is fundamental: What’s an elementary particle? Is it an element of a Hilbert space? Or is it described by an element of a Hilbert space? Or, to put the question differently: Is there anything about reality that cannot be described by mathematics? If you say no to this question, then mathematical objects are just as real as particles.
What Holt actually says is: “I’ve never seen any of the mathematical objects that I’ve heard about in school, thus they don’t exist and Platonism is a phantasy. “ Which is very different from saying “I know that our reality is not fundamentally mathematical.” With that misunderstanding, Holt goes one to explain Platonism by psychology:
“And today’s mathematical Platonists agree. Among the most distinguished of them is Alain Connes, holder of the Chair of Analysis and Geometry at the College de France, who has averred that “there exists, independently of the human mind, a raw and immutable mathematical reality.”… Platomism is understandably seductive to mathematicians. It means that the entities they study are no mere artifacts of the human mind: these entities are discovered, not invented… Many physicists also feel the allure of Plato’s vision.”I don’t know if that’s actually true. Most of the physicists that I asked do not believe that reality is mathematics but rather that reality is described by mathematics. But it’s very possibly the case that the physicists in my sample have a tendency towards phenomenology and model building.
Most of them see mathematics as some sort of model space that is mapped to reality. I argued in this earlier post that this is actually not the case. We never map mathematics to reality. We map a simplified system to a more complicated one, using the language of mathematics. Think of a computer simulation to predict the solar cycle. It’s a map from one system (the computer) to another system (the sun). If you do a calculation on a sheet of paper and produce some numbers that you later match with measurements, you’re likewise mapping one system (your brain) to another (your measurement), not some mathematical world to a real one. Mathematics is just a language that you use, a procedure that adds rigor and has proved useful.
I don’t believe, like Max Tegmark does, that fundamentally the world is mathematics. It seems quite implausible to me that we humans should at this point in our evolution already have come up with the best way to describe nature. I used to refer to this as the “Principle of Finite Imagination”: Just because we cannot imagine it (here: something better than mathematics) doesn’t mean it doesn’t exist. I learned from Holt’s book that my Principle of Finite Imagination is more commonly known as the Philosopher’s Fallacy.
“[T]he philosopher’s fallacy: a tendency to mistake a failure of the imagination for an insight into the way reality has to be.”Though Googling "philopher's fallacy" brings up some different variants, so maybe it's better to stick with my nomenclature.
Anyway, this has been discussed since some thousand years and I have nothing really new to add. But there’s always somebody for whom these thoughts are new, as they once were for me. And so this one is for you.
xkcd: Lucky 10000. |
73 comments:
Dear Sabine
The article on Platonism is quite a nice way to start a discussion on a very difficult issue.
ON the other hand, I think that there is a missing definitional question tha tto be put firsthand before proceeding to judgements about Platonic attitudes.
Whta does Plato really meant by mentioning the "existence" of perfect mathematical objects?
Should we perceived it ontologically we certainly run itno troubles similar to those of the mind/body or matter/spirit dichotomy.
But there is another way to perceive such an existence, more akin to information/symbolic manipulation of nowadays computer science.
If we see maths as an art more than a science, than we may recognise the infinity of psossible math games that an evolutionary higher mind can invent, including their possible limitations as exposed in the work of Goedel.
Thus, re-invention of mathematical activities can be said to be a kind of "invariant" among all intelligent beings that could appear out of evolution in our galaxy or evereywhere in the universe. It is not even necessary to demand that different beings would have the same maths on everyday use. In the opposite corner of our galaxy, some intelligent race could have ended using say, π-adic calculus just becasue it is more convenient for their different senses.
Nevertheless, it is true that in this more broad sense of "potentiality", mathematical activities and their perfect object do trully exist as a capacity of all minds with higher faculties beyond time and space. For me, this is a sufficient and satisfactory explanation for their "existence".
Bee: "It makes perfect sense to think of time as a mathematical object itself"
It doesn't make any sense to me. How is time a mathematical object? Time is a physical process and is quite hard to define but no mathematical object I ever heard of can capture the essence of it's meaning.
Actually it's remarkable how similar mathematical objects used to model time and space are and yet how extremely different time really is from space in reality. An ever changing single instant which cannot be controlled versus a 3D expanse that can be moved through freely.
I'm glad you're more open-minded about Platonism than Holt is - because you understand that many of the initial obvious objections are based on confusion.
I don't think that Platonism says "we humans should at this point in our evolution already have come up with the best way to describe nature." It just says that abstract structures, like mathematical entities, really exist. (Plato called them "forms".)
Personally I like Popper's 'three worlds' scenario where the physical world emerges from the world of mathematics, the world of consciousness emerges from the physical world, and the mathematical world emerges from the world of consciousness.
Penrose drew a cute picture illustrating this idea:
http://cambridgeforecast.files.wordpress.com/2007/05/worldspenrose.gif
Hi PTMR,
You too have a pre-conceived notion of time. Look, here's a definition:
Let tau be a real number.
Call tau time.
Now tau is a real parameter and you can have mathematical objects depend on that parameter.
Whether that has something to do with "physical processes" is another question entirely and one that I didn't touch upon whatsoever. All I wanted to say is that it is perfectly possible to have mathematical objects that depend on time, because that depends on what you mean by time to begin with. Best,
B.
Hi John,
Yes, you're right. I did meander from Platonism to Tegmark's mathematical universe. But let me ask you: if it would turn out that Nature is not described by mathematics but is better described by some other language, would you still believe that mathematical objects are real? You could, but wouldn't the main motivation have gotten lost?
Holt does briefly mention the three worlds scenario and Penrose's take on it. Best,
B.
Hi Bee,
There is this constant need for you to return to the mathematical world and what it means to you? For your Mom too?:)
Not sure of the image John is pointing too...as punching it in did not come up with anything?:)
I think that Fig. 34.1 best expresses my position on this question, where each of three worlds, Platonic-mathematical, physical and mental-has it’s own kind of reality, and where each is (deeply and mysteriously) found in one that precedes it ( the worlds take cyclicly). I like to think that, in a sense the Platonic world may be the most primitive of the three, since mathematics is a kind of necessity, virtually conjuring its very existence through logic alone. Be that as it may, there is a further mystery, or paradox, of the cyclic aspect of these worlds , where each seems to be able to encompass the succeeding one in its entirety, while itself seeming to depend only upon a small part of its predecessor.”(Page 1028-The Road to Reality- Roger Penrose- Borzoi Book, Alfred A. Knoff- 2004)
So the man see's circles. Here's another one Bee.
[ROGER PENROSE]
The following is a quote from Dr. Roger Penrose's closing remarks.....
"One particular thing that struck me... [LAUGHTER]...is the fact that he found it necessary to translate all the results that he had achieved with such methods into
algebraic notation. It struck me particularly, because remember I am told of Newton, when he wrote up his work, it was always exactly the opposite, in that he obtained so much of his results, so many of his results using analytical techniques and because of the general way in which things at that time had to be explained to people, he found it necessary to translate his results into the language of geometry, so his contemporaries could understand him. Well, I guess geometry… [INAUDIBLE] not quite the same topic as to whether one thinks theoretically or analytically, algebraically perhaps. This rule is perhaps touched upon at the beginning of Professor Dirac's talk, and I think it is a very interesting topic." See: Paul Dirac Talk: Projective Geometry, Origin of Quantum Equations
Something to think about anyway eh?:)
Best,
Dear Bee,
"Is there anything about reality that cannot be described by mathematics? If you say no to this question, then mathematical objects are just as real as particles."
You already know this, but perhaps it bears repeating:
If you say no, then **some** mathematical objects are just as real as particles. There is no way, from within mathematics alone, to know which mathematical objects these are.
It is the same problem with language, say, English. It would be very laborious and boring, but all of mathematics can be written down in English. All of what we know of reality can be described in English. That does not prevent us from asking grammatically sound but meaningless questions about mathematics or the world (such as "Why does something exist instead of nothing? :) ). We need the experimental sciences to help us out.
Best,
-Arun
A few more quotes you might enjoy Bee.
"The end he (the artist) strives for is something else than a perfectly executed print. His aim is to depict dreams, ideas, or problems in such a way that other people can observe and consider them." - M.C. Escher
I suppose you are two fathoms deep in mathematics,
and if you are, then God help you, for so am I,
only with this difference,
I stick fast in the mud at the bottom and there I shall remain.
-Charles Darwin
"I’m a Platonist — a follower of Plato — who believes that one didn’t invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered."Harold Scott Macdonald (H. S. M.) Coxeter
I know you might enjoy Prof Dan Shechtman's video and maybe you see something that I see?:)
Best,
And Lastly,
You might think the loss of geometry like the loss of, say, Latin would pass virtually unnoticed. This is the thing about geometry: we no more notice it than we notice the curve of the earth. To most people, geometry is a grade school memory of fumbling with protractors and memorizing the Pythagorean theorem. Yet geometry is everywhere. Coxeter sees it in honeycombs, sun°owers, froth and sponges. It's in the molecules of our food (the spearmint molecule is the exact geometric reaction of the caraway molecule), and in the computer-designed curves of a Mercedes-Benz. Its loss would be immeasurable, especially to the cognoscenti at the Budapest conference, who forfeit the summer sun for the somnolent glow of an overhead projector. They credit Coxeter with rescuing an art form as important as poetry or opera. Without Coxeter's geometry | as without Mozart's symphonies or Shakespeare's plays our culture, our understanding of the universe,would be incomplete. DONALD COXETER: THE MAN WHO SAVED GEOMETRY by SIOBHAN ROBERTS
Best,
Hi Arun,
Yes, you're right, that's what I meant. Sorry for being sloppy. But look, if you believe only some mathematical objects are real, you need some way to decide which ones are and which ones aren't. Can this assignment be a mathematical requirement? How would this ever get you out of the world of mathematics? If it's not, then mathematics is not all you need to describe reality, in which case there is no reason to believe even some mathematical objects are real. You know I don't believe in Tegmark's mathematical universe, but it seems to me indeed the only reasonable conclusion you can come to is that if some mathematical objects exist "for real" then all of them do. This is not to say that this is actually a logical conclusion. I am saying it's reasonable because the alternative - some mathematical stuff is "real" and there's some way this stuff is assigned to "reality" - doesn't make much sense to me. Where is the rest then? Or maybe I should better ask, what is the rest? Best,
B.
The point is that a bunch of physicists were trying to make sense of atoms and were going around in circles and tearing their hair out, until Jordan mentioned Quaternions to Pauli - and then the problem was solved with Spin. Color expands on that. The problems were solved so conclusively that there is no reason to think there is anything else involved than a mathematical pattern. We may have problems interpreting algebra, but algebra does not go away - instead physics gets more and more reliant on it. So why do we get cranked up about fuzzy words when we already know that interpretation is a big hassle ? And it is not 'all of math' anyway. It is the simplest things that are a hassle because we can not inspect electrons directly. If we could, there would be no hassle with Spin, or why the SM needs to put right handed electrons in a singlet and lefties go with doublets. It is an endless discussion because it is difficult to discern what is a convenient mathematical fiction from what Nature is actually up to.
in Misner,Thorne, Wheeler there was a picture of a letter U with an eye -the universe looking at itself. A bit more succinct than Penrose.
"Is there anything about reality that cannot be described by mathematics?" self-awareness, intelligence; to a lesser extent, turbulence. Does Heisenberg imply numbers should be fuzzy rather than discrete? The pendulum equation is deeply fundamental for excluding mass. However, it has an infinite tail of even sign powers of theta. Really? Choose your weapon when sign(theta) does not equal theta (in radians):
http://www.uwstout.edu/mscs/upload/Deterministic-Modeling.ppt
versus
Int. J. Math. Educ. Sci. Tech. 40(8) 1109 (2009)
f(x) = e^[-x^(-2)]
f(0.1) = 3.7×10^(-44)
f(0.1i) = 2.7×10^43
Don't step on a 10^87 crack.
(3,472,073)^7 + (4,627,011)^7 = (4,710,868)^7
Sorry, Fermat. "8^>)
10^500 empirically irrelevant universes and parity violations enlessly patched with manually inserted symmetry breakings suggest spacetime is fundamentally trace parity-odd toward matter. Mathematics is not empirical. Mathematics can model but not repair hubris.
Dear Bee,
I guess all we know is that some mathematics is useful in describing reality.
People like to think that, say, non-Euclidean geometries existed before humans first became convinced of their existence; they always existed, and just had to be discovered. Where did they exist however? In a Platonic world.
Do things spring into existence or vanish out of existence in the Platonic world? The intuition is that they do not, what is in it always existed, and will always exist, and thus the sense of timelessness.
One can meaningfully ask, however, to what extent does a computer programming language like C++ or LISP belong to the Platonic world of mathematics?
a. I reject the idea that programming languages are entirely not mathematical.
b. I reject the idea that C++ or LISP has always existed in some Platonic world, and were simply discovered. These languages are the product of design and choices.
c. If one admits C++ or LISP into the Platonic world, one is in trouble. Why is our notion of integers or topology any more "fundamental" than C++ or LISP? Also, the notion of the timelessness of the Platonic world evaporates.
d. If one does not admit C++ or LISP into the Platonic world, one runs into other troubles. E.g., I believe any algorithm can be expressed in these languages, in as precise a way as in any other notation. One might argue it is the abstract algorithm that belongs to the Platonic world and not any expression of it. Extending that idea, only the abstract notion of "set" belongs to the Platonic world, but no actual instance of set? What about the set of integers?
e. We run into further troubles when we realize that more than two thousand years ago, Panini made the grammar of Sanskrit as precise as that of C++. Sanskrit, however, is a literary, not a mathematical language.
f. Does the Platonic world contain only those mathematical objects about which we have no choice (of the kind Stroustrup had when he designed C++)? It is hard to make that case, because we do choose axioms; e.g., it is a choice that flips us between Euclidean and non-Euclidean geometries.
g. Does answer to any of the above questions advance our knowledge? (if not, why waste time on it?)
Best,
-Arun
Concerning Platonism, I think my views are close to Arun's if I'm reading him right. I mean if Platonism makes some mathematicians feel better about their work, good for them but to me platonic plane of existence is just useless metaphysical fluff.
I can see how there may be some appeal of Platonism for simple universal ideal objects but if one takes this ideology seriously then it should also apply to all other abstract concepts. Arun mentions programming languages, but the same applies to any language, English included. Take "tree" for example, the concept doesn't refer to any particular tree but to an idealized abstract category of objects, Platonism implies that we also did not invent the concept of a tree, it must have been living in a platonic plane of existence and we simply discovered it. I don't see any justifiable criterion which would admit say spheres to the platonic realm but not trees.
But the same goes for hammer, esophagus, episcopal church and every other concept in every other language. So if there is platonic plane of existence it must be filled with all possible categories, forms, meanings and relations. Does attributing existence to all this abstract junk mean anything? Not to me.
There´s a nice podcast and a PDF book on this subject here:
Science's First Mistake: Delusions In Pursuit of Theory?
http://www.gresham.ac.uk/lectures-and-events/sciences-first-mistake-delusions-in-pursuit-of-theory
Their main point is: Mathematical objects are absurd and not always logical. But they are useful anyway.
I like some of the quotes they use:
“In mathematics you don't understand things. You just get used to them.”
John von Neumann
“When one rows, it is not the rowing which moves the ship: rowing is only a magical ceremony by means of which one compels a demon to move the ship.”
Friedrich Nietzsche
I have some methematicians as colleagues but somehow they never want to discuss my questions like:
"Those mathematical objects of yours (rings, groups etc.) ... do you invent them or do you discover them?" "Have they been there already or do you design them?"
One might ponder....as to how all things come from nothing?:)
Best,
/* claiming that mathematics studies "timeless objects" is sloppy and misleading */
The fact the math can handle the time in similar way like with another quantities doesn't change the fact, that concept of time doesn't follow from Peano algebra at all. The math deals with countable objects (natural numbers aka particles) and gradients (differentials). Formal math is atemporal by its very definition. What is valid now in it remains valid for ever without any evolution - which is why the math language is used for communication of ideas, after all.
BTW In this context it's not accidental, the formally thinking people are conservative if not autistic by their nature and they tend to atemporal, schematic thinking often: as a crystalline example may serve Lubos Motl, but Bee tends to similar behavior too. The autists avoid all change of established rules in their life.
If I give you two mathematical problems to solve, "Riemann Hypothesis" or the "Navier–Stokes existence and smoothness" do you know the full use of these applications.....why then is their room left to figure it out? Why offer a prize if there is nothing left to consider.
Best,
/* Is there anything about reality that cannot be described by mathematics? */
Of course, for example every bitmap or letter, which I can see at this page. The point is in effectiveness of description: if the description requires higher volume of information, then the object itself, it's not description anymore.
“Insofar as the propositions of mathematics refer to reality, they are uncertain; insofar as they as they are certain, they do not apply to reality.”
Albert Einstein
When one speaks about time, one can also ask "About what time?". The time of our subjective experience - irreversible and with no future and past represented as memories? Or the geometric time of general relativity and modern physics- reversible and future and past in a similar position?
If mathematical objects depend on time, the understanding /percepts about them- can depend on the subjective time - popping up to the collective mathematical consciousness and gradually polished and refined to more detailed percepts as mathematics evolves.
On physics side quantum states as fundamental building bricks of objective existence looks rather attractive idea. There would be no "real physical" worlds behind their mathematical representations. They could represent the Platonia of mathematician and have a capacity to represent all internally consistent mathematical structures. This is more or less what Max Tegmark assumes. Mathematician is however lacking from this picture.
Maybe mathematician could be brought in as quantum jumps between these Platonic objects. They would give rise to subjective reality -something at different ontological level than "objective realities" of Platonia. Conscious experience as re-creation, replacement of Platonic object with new one would give information about these Platonic objects. "Me" could be seen as a spotlight of attention hopping around Platonia.
Dear Arun,
Sorry, I don't see what trouble you're referring to with c). Do programming languages belong to the world of mathematics? Sure. Why these? Well, every other one is also there, we just don't use them. Or maybe we do, but in some other part of the multiverse. That's not to say that I actually believe that, just that I don't see how it's inconsistent. Best,
B.
Hi PTMR,
Yes, sure, all that is there too, and many other things. As long as you can define them some way. Best,
B.
Independently of Platonic existence, I might mention that a 'thing' is quite real in the sense that if a person believes in a thing, then it exists damn near well enough for them to act on their conception of it.
Another angle is that Mathematics is the manifestation of logic in the physical theories. It's the tool of logic if you like, which when used in the construction of physical theories, forbids any irrational constructions.
Well, all good mathematicians should first know what is a number. Penrose maybe saw the difficulty when he with every number added an unknown imaginary number, so every number would be complex?
This may explain the phenomenon that the sum is bigger than its part in the holographic principle? The numbers are treelike structures say Susskind. Or booklike as Matti Pitkänen says.
The quantum artithmetic is also badly explored.
Bee,
Long time since I last commented but amazingly, this is the first time I've actually thought seriously about this question!
I've since switched to a masters in applied mathematics and so my viewpoint is probably firmly entrenched in model building, since that is what I do.
However it still amuses me that I haven't actually thought about this before in any serious. Although someone above mentioned whether or not mathematics is discovered or invented, a subject I sometimes think about but soon get bored -because I have no idea- and continue on.
Unlike the philosophy of science or quantum mechanics (both of which I have studied in a formal sense) I've never gotten around to the philosophy of mathematics.
Well wherever I stand, I certainly will have to think about it more.
Dear Bee,
With (c) I was anticipating the objection that somehow integers are universal and C++ is not. E.g., every human culture has learned how to count; but only very recent ones have C++. And who knows, C++ may eventually go extinct too. So then only universal things go into the Platonic world.
Best,
-Arun
Did Holt really write "phantasy" instead of "fantasy"? (The "ph" form is valid, but obsolete, English.)
No, he didn't. That was my mistake when typing off his text, spell-check didn't catch it. Thanks for pointing out, I've fixed that.
Mathematics: tactics of self-referential systems (e.g. the human brain) to mimic "larger" self-referential systems (from which the former are part of, namely, of nature). The tactics by itself *is* a self-referencing system— in fact, the only one which cannot entirely mimic itself (by Gödel's theorem), because it is the "larger", ultimate, self-referencing system possible.
@ Christine.
The set of the integers themselves if represented as a matrix of polynomial coefficients (coefficient -> color) is self-referential and it does not seem to suffer from Cantor's diagonal argument due to its internal self-similarity. Is this evidence tha tno more than the integers are necessary for building toy universes?
http://cag.dat.demokritos.gr/Lexicons.php
http://cag.dat.demokritos.gr/Observer.php
Theophanes,
Maybe. In any case I used the word "entirely" so toy universes were not on my mind.
Best,
Christine
Alain Connes made a very enigmatic statement about Platonism I do not understand:
"... the point is the existence of true - not just undecidable
- but unprovable statements. Unless you understand that point it is worthless to debate about Platonism."
See here: http://www.math.ru.nl/~landsman/ConnesNAW.pdf
Does anybody have an idea what he means by that ?
There may be something of value in this blog entry?
Conversations on Mind, Matter, and Mathematics with regard to Jean-Pierre Changeux and Alain Connes
The Book:
Do numbers and the other objects of mathematics enjoy a timeless existence independent of human minds, or are they the products of cerebral invention? Do we discover them, as Plato supposed and many others have believed since, or do we construct them? Does mathematics constitute a universal language that in principle would permit human beings to communicate with extraterrestrial civilizations elsewhere in the universe, or is it merely an earthly language that owes its accidental existence to the peculiar evolution of neuronal networks in our brains? Does the physical world actually obey mathematical laws, or does it seem to conform to them simply because physicists have increasingly been able to make mathematical sense of it? Jean-Pierre Changeux, an internationally renowned neurobiologist, and Alain Connes, one of the most eminent living mathematicians, find themselves deeply divided by these questions.
Best,
Joel,
... the point is the existence of true - not just undecidable
- but unprovable statements. Unless you understand that point it is worthless to debate about Platonism."
With regard to Alain Connes statement, could it possibly mean that once you get to "this point" where do you draw from in experience? Was it written somewhere in nothing land?
Where a dictionary proceeds in a circular manner, defning a word by reference to another, the basic concepts of mathematics are infinitely closer to an indecomposable element", a kind of elementary particle" of thought with a minimal amount of ambiguity in their definition.Alain Connes
So it is an exercise in logic reduction to "make available" an opportunity to the resources from which all mathematical information can be drawn?
It's out there somewhere, not in nothing.
Best,
Joel,
He is referring in part to Gödel's theorems, but why he relates those to Platonism is not clear. Maybe you should check on the book by Jean-
Yves Girard, cited by him.
Best,
Christine
Christine,
you're right, one should try to find the answer in the book.
I had already a quick look at it but couldn't find the respective passage.
It seems to me that it is a very unconventional book about logic and to understand it one has to read it from the beginning. Hope I'll find the time.
Maybe this video is also helpful:
http://www.canal-u.tv/video/universite_de_tous_les_savoirs/les_fondements_des_mathematiques.1021
Plato Hagel,
I think your comments point in the right direction.
Finally I found this:
http://books.google.de/books?id=xiS9y8SLq0wC&printsec=frontcover&hl=de&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
On p. 5-7 A.C. explains in more detail the difference between mathematical "truth" and provability.
I wonder if this is related to the issue of "non-computability" of the human mind, put forward by Roger Penrose. Is this why we humans can do mathematics whereas a computer cannot ?
Joel
See also this link:
http://guidetoreality.blogspot.com.br/2006/12/gdels-platonism.html
Best,
Christine
Joel:I wonder if this is related to the issue of "non-computability" of the human mind, put forward by Roger Penrose. Is this why we humans can do mathematics whereas a computer cannot ?
There are some interesting quotes here in following article that come real close to what is implied by that difference.
You raised a question that has always been a troubling one for me. On a general level how could such views have been arrived at that would allow one to access such a mathematical world?
The idea being that to get to the truth one had to turn inside and find the very roots of all thought in some geometrical form. The closer to that truth, the very understanding and schematics drawn in that form. Not all can say the search for such truth resides within? Why the need for such geometry in relativism? Riemann Hypothesis as a function of reality? Why has a computer not solved it?
My views were always general as to what we may have hoped to create in some kind of machine or mechanism. I just couldn't see this functionality in relation to the human brain as 1's and 0's.
I might say it never occurred to me the depth that it has occupied Penrose's Mind. The start of your question and the related perspectives of the authors revealed in the following discourse have raised a wide impact of views that seek to exemplify what is new to me as to what you are asking.
Yet the real world has made major advancements in terms of digital physics and hyper physics. Has any of this touched the the nature of consciousness. This would then lead to Penrose angle in relation to what consciousness is capable of and what a machine is capable of. That would be my guess.
Thanks for the question.
Best,
Christine,
thanks for the link.
It see great similarities in the thinkings of some of the deepest thinkers like Gödel, Von Neumann, Penrose, Connes etc., saying sth. like there had to be a reality/truth beyond finite formal systems, the latter only approximating it.
I think one must take them very serious (if one likes this idea or not) and it should be worth further ponderings!
Gödel was a very tragic figure, it seems. If you haven't seen this video, I highly recommend it:
http://lockerz.com/u/20572546/decalz/6442529/dangerous_knowledge_g_cantor_l_boltz
Best
You raise several questions, that circulate in by brain since many years and where I do not have a (definite) answer.
I only recently converted to a Platonist due to the following experiment:
http://www.univie.ac.at/qfp/publications3/pdffiles/Xiaosong_Experimental%20delayed-choice%20entangelment%20swapping.pdf
The only way I can make sense out of it is to assume that quantum physics is a mathematical reality beyond space and time. If this is so, the step from physics to Platonism is not a too big one for me any more.
There is an interesting conclusion at the end of the paper:
"If one viewed the quantum state as a real physical object, one could
get the paradoxical situation that future actions seem to have an
influence on past and already irrevocably recorded events. However,
there is never a paradox if the quantum state is viewed as no more
than a 'catalogue of our knowledge'. Then the state is a probability
list for all possible measurement outcomes, the relative temporal
order of the three observers' events is irrelevant and no physical
interactions whatsoever between these events, especially into the
past, are necessary to explain the delayed-choice entanglement
swapping."
Maybe this helps to shed some light on your question:
"One might ponder....as to how all things come from nothing?:)"
Best
Further thoughts perhaps.....
Computational Dilemma
“The only way I can make sense out of it is to assume that quantum physics is a mathematical reality beyond space and time. If this is so, the step from physics to Platonism is not a too big one for me any more.”
Not surprisingly, this does not follow for me. It would seem a matter of mental hygiene for one to be able to distinguish between what is real and what is an artifact of cognitive abstraction. In general one could make that distinction by saying that real things have a kind of pulse, an energetic regimen, the amplitude of which is negatively correlated to the level of abstraction.
I was hoping the applicability of the 2nd law of thermodynamics would serve a litmus test for distinguishing between the real and the Platonic events, but apparently it is less of a clear beacon at the quantum level.
In any case, nature has normative pathways in which energy makes transitions and we map them as the laws of physics. Even if we can only characterize these pathways via probabilities surely there remains a distinction between those pathways and our mapping of them.
Of course, I could well be missing the deeper discussion here.
The issue is of where the information exits, if it can exist at all.
Planck length measure reveals the issue of "what geometry" can explain it....yet such a foundation asks that the Higg's mass is defining here as to what details the matter assigned to it? Where did such schematics come from?
Best,
Tegmark believes in an extreme form of Platonism, the idea that mathematical objects exist in a sort of universe of their own. Imagine that, Tegmark says, “there’s this museum in this Platonic math space that has these mathematical objects that exists outside of space and time,” Tegmark says. “What I’m saying is that their existence is exactly the same as a physical existence, and our universe is one of these guys in the museum.”Wigner’s Gift Horse By JULIE REHMEYER • Feb 1, 2008 See here for article.
Bateson offered the tidy definition of information as "a difference which makes a difference." If we make that “measurable difference” will that serve to roughly characterize our sense of physical information?
Further, if can we accept as a first axiom David Bohm’s notion that the universe is one, undivided and entire, then it follows that any “difference” would arise topologically, be in the nature of a fold rather than in some distinction of a fundamental nature.
If we can imagine, as Tegmark suggests, that, “there’s this museum in this Platonic math space that has these mathematical objects that exists outside of space and time,” may we also imagine a mythical universe that creates its first difference by somehow folding upon itself?
Within our imaginary realm, we could refer to these two, folded manifestations of the whole as the Traveler and the Terrain. We might find that they are endlessly entwined but never consummate in touch because they are star-crossed, laid topologically cross-grained, one to the other.
We can never know either of them directly, but only the paths that emerge between them. This is because we live nested within the entwinement of these paths and they are all we are given to know.
We find that if we follow a path that approaches too close to either the Traveler or the Terrain, then part of our measurement of that path dilates, spreads across the surface of its compliment and becomes indeterminate.
Nevertheless, we are able to study the nature of many paths and find that they exhibit similar iconic forms Can we say more than this or is it simply the nature of things in our imaginary universe?
Don,
"Not surprisingly, this does not follow for me. It would seem a matter of mental hygiene for one to be able to distinguish between what is real and what is an artifact of cognitive abstraction."
So then, what do you think about people doing quantum cosmology, assuming that spacetime comes about through a quantum fluctuation, implicitly assuming that quantum physics is already there but not spacetime.
To get the idea:
http://www.youtube.com/watch?v=yieI58joOkM
Best
Plato Hagel,
"...yet such a foundation asks that the Higg's mass is defining here"
The Higgs is yet another gauge symmetry, yet one of a discrete noncommutative space. (Thus again, GEOMETRY).
For details, see Conne's noncommutative standard model.
Best
Thanks Joel,
Alexander Vilekin is well groomed and gives the impression of having a very tidy mind, like someone it would be a pleasure to talk with, ask questions of. He makes an easy to follow, plausible case for the universe arising from some grand quantum fluctuation. I appreciate physics at the storybook level.
Still, considering his metaphor of a bubble, there is a tension implicit in that image, the impression of an expansion dynamically mediated by a constraint. What is the nature of this tension and how does it arise?
Would you consider it possible that, preparatory to the emergence of space-time, we might find the dynamic frisson of quantum fluctuations arising between the “surfaces” of some underlying topological polarity?
Regards, Don
Hi Joel
I found this suitable.
Is Our Intuition Lost?
What is the price one pays for dealing with Noncommutative Geometry? Probably the most significant one is that the concept of space (understood as a set of points) is lost and can no longer visualize the objects we are dealing with. This is not new in physics: quantum mechanics has taught us to deal with the phase space, which cannot be imagined as a manifold. However, since here one wants to do geometry this lost intuition is somehow worrying. What is closely connected with this point is that there is no obvious guiding principle, which would tell us which objects are "good" or "relevant" - in the "spatial" geometry the intuition was often a key, here, Noncommutative Geometry is like entering a new, partially unknown land. Of course, does this suggest that we are in need of a new intuition? Why Noncommutative Geometry?
Best,
Hi Plato,
I think you brought up a particularly good point. Intuition plays an important part in physics because it is a subject that deals with the intersection of abstract ideas and mathematics. But you still need to be able to visualize in your mind what you are working on even if you cannot see with the five senses what you are working on. Many scientists have become disconnected from the intuition required to make progress in physics. Many of these people don't even know they have become disconnected from intuition and it shows up in wild suppositions disconnected from physical data. Many of the physicists that have made the biggest advances in the past were working closely with experimental evidence that gently prodded them from going off track. They were lulled into thinking they had innately good intuition because they were kicked when their analysis diagreed with the data.
Now many, not all, of these award winning physicists have gone off track and led the rest of the community with them. It is because they were overconfident and overestimated their intuition based on previous success when they had a lot of phenomenological data that gave them good feedback.
I think noncommutative geometry and how one visualizes it, if one can, is a good measuring stick for the quality of ones innate physical intuition. It really relates to measurement and to the energy required to do those measurements in relation to the energy of the particle or field one is measuring. If the object or field one is measuring is overwhelmed by the energy required to measure it then one has a non commutative geometry involved. It is not a difficult concept and many of us can intuit it directly. I would say if one can't see that without working at it much then one's natural physical intuition is wanting. This visualization will always require the strict observance of the first law of thermodynamics for the universe as a whole. Again, something people with poor intuition throw out way too often. Physicist pay way too much attention to the second law of thermodynamics and not enough to the first. Just my opinion of course.:-)
Claro.
Thanks.
One further thought on noncommutative geometry and why it is associated with tiny spaces and objects only. Gravitationally significant objects occupy a significant volume of space. This is because the G force is ~ 10^-42 smaller than the strong force. You can measure most effects of gravity without deforming the space around it by measuring it.
There is no such thing happening with the three other forces. To get any good local knowledge of any of their fields one must physically deform the field just to measure it. However, when you measure gravity the energy required to measure it (not quantum gravity) is infinitesimal compared to the object you are measuring. Thus the gravitational field is not deformed during measurement and is a commutative measurement, I.e. It behaves classically.
Plato,
very nice statement. I fully agree.
In terms of intuition, I have nothing to add to this:
http://www.youtube.com/watch?v=jrk3GbJU0k0&feature=related
Regards
Don,
I share your scepticism and can't give you an answer to your question.
Dirac once said:
"One field of work in which there has been too much speculation is cosmology. There are very few hard facts to go on but theoretical workers have been busy constructing various models for the universe based on any assumptions that they fancy. These models are probably all wrong."
Given our progress in experimental cosmology since, maybe one should replace "cosmology" by "quantum cosmology" these days. :-)
Regards
Joel,
The Feynman clips are delightful. Also enjoyed this one in which he notes that features of our individual mental landscapes and processes may vary significantly and require some translation:
http://www.youtube.com/watch?v=lr8sVailoLw&feature=related
In the clip on intuition he clearly make the case that there are realms, which are only accurately accessible through mathematics.
Still, I wonder if features of quantum processes manage to eddy upward to the tangible surface of everyday experience. It’s a matter observing it. After all, John Scott Russell first observed the dynamics of the soliton while on horseback.
Regards, Don
Joel and Don,
Reading and hearing about Richard Feynman is what got me Interested in physics 25 years ago. Particularly his thoughts on the quantum world. It was just too fantastic to not be fascinating. Also, I concurred with his belief that the quantum world is unvisualizable.
However, as charismatic as he was and is he was only a human being. A human being with his own prejudices and preconceived notions. In other words, my thoughts on the quantum world has evolved over the 25 years when I first read about the quantum world through reading about Feynman. It is best not to get locked into one mans view of how to look at the world, or a portion of the world.
"It is best not to get locked into one mans view of how to look at the world, or a portion of the world."
Make that ANY one man's view, most definitely including my view of how things work.
Eric,
"It is best not to get locked into one mans view of how to look at the world, or a portion of the world."
Of course, we are all human, we are all limited.
Here is a very nice illustration that demonstrates the "tension" of our natural intuition and the quantum "reality":
http://plus.maths.org/content/proof-kochen-specker-theorem
Best
... which by the way proofs that Einstein, another genius was "wrong" ...
Dear all
Tegmark is 100% correct because my theory "quantum statistical automata" proves it.
Reality exists hence we say it is true. But what is really true besides that more than anything else which we can really trust, it is mathematical facts. So, to my mind I connect both since both seem to be a statement of truth. So I took a guess that reality is something akin to a circle (truth). The relations between the points give you a mathematical structure whereby you get PI which defines the structure of the circle.
the structure that leads to our reality is random numbers and certain unavoidable relations(and only possible ones) between them. that is all.It is the most generalized structure possible. The mass of the electron pops from the system among many other standard physics results.
Once you understand it, it will become clear to you that it could not have been otherwise.
Qsa,
very interesting. I am not shure I understand what you are doing.
"Set a constraint with a particular relation between "l", "p", and "li" such that if this relation holds.
What determines "A CONSTRAINT" ?
Couldn't it be that you have just come up with the right Monte Carlo algorithm for the discretized Schrödinger equation.
In other words, changing the constraint you could emulate any equation?
It would be interesting to see how to get solutions to the Dirac equation this way.
Best
Joel
Thank you for your interest. Your Question about the constraint is at the heart of the system. The system (I will not use the word model although you could with some caution) seems to mimic reality by exposing some of the very important essential features of Schrodinger equation, Dirac equation, QFT and Gravity! But only certain essential features of these theories, probably some heavy work and more elaborate simulations needed to map to the of standard physics. I will elaborate about the constraints a bit later.
On the other hand the system exposes features of reality that standard physics is simply in no position to do so. Particularly, the Lagrangian of the system falls out from the simulation and you get the values of charge, mass, c, h_bar and other values, even the Fine Structure Constant. Not to mention the beautiful unified picture of space (its points are the crossing of the lines-dynamic-), time(change of state-does not actually exist-), mass, charge, and energy.
It is just the nature of the system that suggests to you right away this constraint that I use to mimic the Schrodinger equation and in other situation and gives good results. To be honest I don’t know why it works and that is going to be the subject of investigation. Moreover, another even more useful constraint I use (the simplest) goes to produce all the results of QFT and that is to assume a particle width with uniform random distribution( also should be subject of investigation). This produces the 1/r law with Zee in his book calls the triumph of the 20th century physics. Also many other results listed in the website, and some other staggering results concerning the proton that I have not documented.
The other really big result which I obtain is the essence of Dirac equation included the notorious non-locality. When I try to simulate the 2D situation, I am forced to restrict my line throwing activity to only lines that can go between particles directly so as to keep the invariance of quantities calculated in case the frame is rotated. And Wala, I get two particles to interact through their width in the second axis and it does not matter if each is on the other side of the universe, they are both linked!!!! When I calculate spin (what I believe to be) one is up the other down.
There are many other things in the system which I have not tried to do too much yet mainly due to lack of time including gravity which I have done in limited way and I do get the small attractive force but probably much more work is needed. I have answered you question regarding “any function” at length in the website but I will elaborate if you insist.
I should mention that the theory is at heart a generalization of Buffon’s needle which ties to many other concepts which are used in high end physics like twisters and others.
http://en.wikipedia.org/wiki/Integral_geometry
http://en.wikipedia.org/wiki/Buffon's_needle
You can email for more information and I will help anyone who wants to run those programs and maybe you can come up with your own, it is relatively easy.
Joel
I like to add an amendment to my last post. You compatriot Dr. Christoph Schiller (you can find him at motionmountain) has come up with a theory that essentially boils down to mine. His is a sort of phenomenological in nature (I don’t agree with everything he says), mine is more concrete. Still the two pictures are complimentary. Also, my model has also a hint of the standard model but I leave that for later discussions. Also while the system looks instantaneous and action at distance but it also carries a non-instantaneous signal to express the interaction event at the same time.
Hi Joel,
Getting back to your thesis about not being able to depend on intuition for QFT. You are hiding an assumption that you haven't recognized. You are assuming everyone's intuition is equal. This was the hidden assumption that Feynman overlooked. He was a mathematically oriented physicist and was biased to the toolkit he had in his possession, mathematics. Many others that are mathematically oriented do the same.
This is another case of confirmation bias where mathematically, highly trained physicists cannot believe that there is not another way of "seeing". I would argue that Einstein earlier in his career was using his intuition to guide to guide him and using mathematics merely to corroborate that intuition. He was not wrong in doing that and I'm confident his theory of gravitation will eventually be just the limiting case in a broader theory that combines both quantum and classical aspects. Don't depreciate people with good intuition just because mathematics isn't their primary thing. I certainly don't depreciate people with better mathematics skills than I have just because that is not my gift.
Like I said, believing totally in Feynman's non visual aspect of qft is an example of confirmation bias.
Qsa, Eric,
thanks for your explicit responses. I have to think about what you are saying but am currently just lacking time. Hope to come back to this blog soon.
Best
Eric,
you may be right, it is a matter of bias. But being a trained mathematician and physicist does it then make sense to discuss this matter at all with me ? Yes I do believe that mathematics is crucial in understanding the world, and I would like to give you some great quotes that stress this, but unfortunately they are by people in the field (you see I am biased) and so, would you give any credit to them?
"I would argue that Einstein earlier in his career was using his intuition to guide to guide him and using mathematics merely to corroborate that intuition."
Yes, that is exactly how science works. First the intuition, the idea and then making it rigorous by means of the tools of trade of mathematics ("shut up an calculate" :-)). Actually most of the ideas (even seemingly beautiful ones) fail to work out (a remark by Wilczek - Nobel price winner - by the way).
Best
I think that mathematics being fundamental to reality is virtually beyond dispute. There are countless examples of pure mathematical discoveries being linked to reality decades or centuries later. In a recent post on this blog, the article mentions the aharonov-bohm effect, which seems to show that potential fields (seemingly mathematical constructions) are in some essence real. And those that suggest it needs to be in a platonic realm are perhaps making things more difficult than they need to be. Our universe is, as Tegmark suggests, the mathematical world. I want the physicists to seriously question how they can believe in point particles, which may be made of tiny vibrating strings of energy (well what the hell is that exactly), which seem to boil down to abstract mathematical relations at the bottom, and still hold on to the fact that at some deep level the universe itself doesn't simply boil down to math and logic. I want to stress that this doesn't need to be mystical or religious at all, but I feel like some scientist's get scared because of the religious overtones the word "Platonism" induces. If humans didn't exist, the fundamental relationships that mathematics reveals would obviously still be real.
Hi Pete,
As a supporter of the mathematical universe hypothesis, is it possible to look at my theory a bit in more detail and give me your opinion. I cannot give you a direct link since Bee is against that but I will give you my email in case you are interested. Thanks.
my email
is sadeq.adel985@gmail.com
“I think that mathematics being fundamental to reality is virtually beyond dispute.”
“If humans didn't exist, the fundamental relationships that mathematics reveals would obviously still be real.”
Perhaps it is my flat-footed lack of imagination, but the notion that mathematics is primal seems akin to putting the cart before the horse. More formally, it is an example of the fallacy of affirming the consequent.
Just because we are in awe of our mathematics’ ability to mirror the dynamics of the real world it does not necessarily follow that the real world is driven by those mathematics. While the real world is by its very nature “energos” (active, at work), mathematics is animated by the human mind. Or, so it seems to me.
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